#6802812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,21,25,22,26,35,41)(19,27,29,37,30,40,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14);(3,8)(5,7)(10,13)(14,15)(17,19,22,30)(18,23,26,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(25,40,41,27)(33,48,38,45)(39,43,47,42);(2,6)(5,7)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,39)(26,31)(27,42)(29,45)(30,44)(34,48)(35,38)(37,46)(40,43)(41,47);(9,11,12,16)(10,14,13,15)(20,32)(24,31)(28,44)(33,38)(36,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6710008);