#6685302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,30)(24,38)(25,34)(26,36)(28,37)(29,41)(31,35)(33,39)(40,43)(42,44);(1,2)(3,4)(6,9)(7,11)(13,15,19,26)(14,20,23,36)(16,24,29,40)(17,31,18,25)(21,33,37,44)(22,35,32,34)(27,38,41,43)(28,42,30,39);(2,4)(13,16,18,30,19,29,17,28)(14,21,22,27,23,37,32,41)(15,24,25,39,26,40,31,42)(20,33,35,38,36,44,34,43);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,29,30)(20,34,36,35)(21,27,37,41)(24,39,40,42)(33,43,44,38);(2,4)(5,7,8,11)(6,10,9,12)(16,29)(21,37)(24,40)(27,41)(28,30)(33,44)(38,43)(39,42)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592498);