#7006205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913401);