#7145758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

2190 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)(3,4)(7,11)(10,12)(13,14,20,32,19,17,18,23)(15,24,33,21,27,37,26,31)(16,22,34,42,29,28,30,36)(25,38,43,35,39,44,40,41);(6,9)(7,11)(13,15,19,27)(14,21,17,31)(16,25,29,39)(18,33,20,26)(22,35,28,41)(23,37,32,24)(30,43,34,40)(36,44,42,38);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,25)(17,28)(18,30)(19,29)(20,34)(21,35)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(33,43)(37,44);(6,9)(10,12)(13,17,20,23,19,14,18,32)(15,24,33,21,27,37,26,31)(16,28,34,36,29,22,30,42)(25,38,43,35,39,44,40,41);(5,7,8,11)(6,10,9,12)(16,29)(22,28)(25,39)(30,34)(35,41)(36,42)(38,44)(40,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052954);