#6574737

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

Polycyclic presentation:

2^9|a^2=fi,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, 6481933);