#6667945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575141);