#6815026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6722222);