#6817222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6724418);