#6685656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1887 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,24,33)(16,20,28,37)(17,21,31,40)(23,32,41,43)(25,34,26,35)(27,36,42,44)(29,38,30,39);(6,9)(7,11)(13,15,18,24)(14,19,22,33)(16,27,28,42)(17,25,31,26)(20,36,37,44)(21,34,40,35)(23,30,41,29)(32,39,43,38);(1,2,3,4)(13,16,18,28)(14,20,22,37)(15,23,24,41)(17,30,31,29)(19,32,33,43)(21,39,40,38)(25,27,26,42)(34,36,35,44);(6,9)(10,12)(15,24)(16,28)(19,33)(20,37)(25,26)(29,30)(34,35)(38,39);(5,7,8,11)(6,10,9,12)(15,25)(17,31)(19,34)(21,40)(23,27)(24,26)(29,30)(32,36)(33,35)(38,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6592852);