#6815900

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2137 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)(13,14,19,24)(15,20,27,36)(16,21,26,35)(17,30,32,22)(18,23,28,37)(25,41,43,34)(29,39,31,38)(33,44,42,40);(6,9)(7,11)(13,15,19,27)(14,20,24,36)(16,28,26,18)(17,31,32,29)(21,37,35,23)(22,39,30,38)(25,42,43,33)(34,44,41,40);(1,2,3,4)(13,16,18,15,19,26,28,27)(14,21,23,20,24,35,37,36)(17,31,33,43,32,29,42,25)(22,39,40,41,30,38,44,34);(6,9)(10,12)(17,32)(22,30)(25,43)(29,31)(33,42)(34,41)(38,39)(40,44);(5,7,8,11)(6,10,9,12)(13,17,18,33,19,32,28,42)(14,22,23,40,24,30,37,44)(15,25,26,31,27,43,16,29)(20,34,35,39,36,41,21,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6723096);