#6814579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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
prime radical	C₂×#3204	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(15,20)(16,26)(17,22)(18,32)(19,23)(21,28)(24,35)(25,33)(27,39)(29,40)(30,36)(31,37)(34,41)(38,44)(42,43);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,30,31,25)(21,26,35,40)(22,36,37,33)(27,41,42,38)(34,43,44,39);(1,2,4,3)(13,16,15,24,19,29,18,28)(14,21,32,40,23,35,20,26)(17,27,25,38,31,42,30,41)(22,34,36,43,37,44,33,39);(6,10)(9,11)(13,17,19,31)(14,22,23,37)(15,25,18,30)(16,27,29,42)(20,33,32,36)(21,34,35,44)(24,38,28,41)(26,39,40,43);(1,3,4,2)(7,12)(9,11)(17,31)(22,37)(25,30)(27,42)(33,36)(34,44)(38,41)(39,43)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=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, 6721775);