#6814927

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

1877 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,26,34)(16,21,28,36)(17,22,31,39)(18,23,32,40)(25,33,42,44)(27,35,29,37)(30,38,41,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,24,34)(16,27,28,29)(17,18,31,32)(21,35,36,37)(22,23,39,40)(25,41,42,30)(33,43,44,38);(1,2,3,4)(13,16,18,30,19,28,32,41)(14,21,23,38,24,36,40,43)(15,25,17,27,26,42,31,29)(20,33,22,35,34,44,39,37);(6,10)(9,11)(16,28)(21,36)(25,42)(27,29)(30,41)(33,44)(35,37)(38,43);(7,12)(9,11)(13,17,18,26,19,31,32,15)(14,22,23,34,24,39,40,20)(16,29,30,25,28,27,41,42)(21,37,38,33,36,35,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=h,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, 6722123);