#72322284

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#33520	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bgi,c^a=cj,d^a=dh,e^a=egh,e^c=ej,e^d=ej,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322319, #72322909
maximal normal subgroup of	#72322319, #72322909

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