#6574721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481917);