#6814902

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

1880 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)(15,26,29,20)(16,30,34,21)(17,25,35,22)(18,23,28,37)(27,36,44,42)(31,38,33,41)(32,39,43,40);(5,7)(6,9)(8,12)(10,11)(13,15,19,29)(14,20,24,26)(16,31,34,33)(17,28,35,18)(21,38,30,41)(22,37,25,23)(27,43,44,32)(36,39,42,40);(1,2,3,4)(13,16,17,32,19,34,35,43)(14,21,25,40,24,30,22,39)(15,27,18,31,29,44,28,33)(20,36,37,38,26,42,23,41);(6,10)(9,11)(13,17,19,35)(14,22,24,25)(15,28,29,18)(16,32,34,43)(20,37,26,23)(21,39,30,40)(27,33,44,31)(36,38,42,41);(7,12)(9,11)(13,18,35,15,19,28,17,29)(14,23,22,26,24,37,25,20)(16,33,43,44,34,31,32,27)(21,38,39,36,30,41,40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6722098);