#72284544

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2403
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	#13867
prime radical	#7708257	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,9)(5,10)(6,8)(7,13)(12,16)(14,15)(17,18,22,23)(19,24,30,29)(20,31,34,25)(21,26,36,37)(27,38,32,39)(28,35,41,43)(33,44,47,40)(42,48,45,46);(1,3,4,13,6,12,5,14)(2,7,9,16,8,15,10,11)(17,19)(18,24)(20,32,34,27)(21,35,36,43)(22,30)(23,29)(25,39,31,38)(26,41,37,28)(33,46)(40,42)(44,45)(47,48);(2,8)(7,15)(9,10)(11,16)(17,20)(18,25)(19,27)(21,33)(22,34)(23,31)(24,38)(26,40)(28,42)(29,39)(30,32)(35,48)(36,47)(37,44)(41,45)(43,46);(2,8)(3,12)(4,5)(7,15)(17,21)(18,26)(19,28)(20,33)(22,36)(23,37)(24,35)(25,40)(27,42)(29,43)(30,41)(31,44)(32,45)(34,47)(38,48)(39,46)

Polycyclic presentation:

2^10|a^2=j,b^2=e,e^2=ij,f^2=ij,g^2=ij,h^2=j,b^a=bg,c^a=cefij,c^b=cefj,d^a=dij,d^b=dh,e^c=ej,e^d=ei,f^b=fj,f^c=fj,f^d=fi,g^a=gij,g^b=gij,g^c=gj,g^d=gij,h^a=hij,h^b=hij,h^d=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284626, #72284997, #72287165, …
maximal normal subgroup of	#72284626, #72284997, #72287165, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);