#72316754

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=e,d^2=i,f^2=ij,g^2=j,h^2=ij,b^a=bf,c^a=cg,c^b=cd,d^a=di,d^b=di,e^a=ej,e^b=ei,f^b=fij,f^c=fij,g^b=gj,h^b=hij,h^c=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316760, #72316910, #72317025
maximal normal subgroup of	#72316760, #72316910, #72317025

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.8);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);