#6667913

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

2773 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)(15,20)(16,26)(17,22)(18,31)(19,23)(21,24)(25,34)(27,40)(28,33)(29,37)(30,38)(32,36)(35,39)(41,44)(42,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,25,30,29)(21,33,36,26)(22,34,38,37)(27,39,42,41)(35,43,44,40);(2,4)(13,16,19,28)(14,21,23,36)(15,24,18,32)(17,27,30,42)(20,33,31,26)(22,35,38,44)(25,39,29,41)(34,43,37,40);(6,10)(9,11)(13,17)(14,22)(15,25)(16,27)(18,29)(19,30)(20,34)(21,35)(23,38)(24,39)(26,40)(28,42)(31,37)(32,41)(33,43)(36,44);(2,4)(7,12)(9,11)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,29,30,25)(21,33,36,26)(22,37,38,34)(27,41,42,39)(35,40,44,43)

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=efh,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, 6575109);