#6814932

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,e^2=hi,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, 6722128);