#6816138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

2156 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)(13,14,19,18)(15,20,25,34)(16,21,27,29)(17,22,33,32)(23,26,35,41)(24,30,37,44)(28,36,31,38)(39,42,40,43);(6,9)(7,11)(13,15,19,25)(14,20,18,34)(16,26,27,41)(17,30,33,44)(21,35,29,23)(22,37,32,24)(28,40,31,39)(36,43,38,42);(1,2,3,4)(13,16,18,29,19,27,14,21)(15,23,20,26,25,35,34,41)(17,31,32,36,33,28,22,38)(24,40,30,43,37,39,44,42);(6,9)(10,12)(16,27)(21,29)(23,35)(26,41)(28,31)(36,38)(39,40)(42,43);(5,7,8,11)(6,10,9,12)(13,17,14,22,19,33,18,32)(15,24,34,44,25,37,20,30)(16,28,21,36,27,31,29,38)(23,39,41,43,35,40,26,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6723334);