#6574689

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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)(16,27)(17,22)(18,31)(19,23)(21,28)(24,39)(25,34)(26,35)(29,37)(30,36)(32,38)(33,42)(40,43)(41,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,20,23,35)(16,24,29,41)(17,30,18,25)(21,33,38,43)(22,36,31,34)(27,39,37,44)(28,42,32,40);(2,4)(13,16,19,29)(14,21,23,38)(15,24,26,41)(17,28,18,32)(20,33,35,43)(22,37,31,27)(25,40,30,42)(34,39,36,44);(1,2)(3,4)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,35,36)(21,37,38,27)(24,40,41,42)(33,39,43,44);(2,4)(15,26)(20,35)(24,41)(25,30)(33,43)(34,36)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,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, 6481885);