#6810576

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6717772);