#6583340

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(18,32)(19,23)(21,28)(24,39)(25,34)(26,35)(29,37)(30,36)(31,38)(33,41)(40,43)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,20,23,35)(16,24,29,42)(17,30,18,25)(21,33,38,43)(22,36,32,34)(27,39,37,44)(28,41,31,40);(2,4)(13,16,18,28,19,29,17,31)(14,21,22,27,23,38,32,37)(15,24,25,41,26,42,30,40)(20,33,36,39,35,43,34,44);(1,2)(3,4)(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,35,36)(21,37,38,27)(24,40,42,41)(33,39,43,44);(2,4)(15,26)(16,29)(20,35)(21,38)(25,30)(27,37)(28,31)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,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=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490536);