#6757327

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 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)(9,10)(11,14)(12,13)(15,16)(17,18,24,28,23,29,22,39)(19,30,35,46,34,25,33,42)(20,27,36,44,21,37,38,26)(31,45,47,40,32,41,48,43);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,34)(18,25,29,30)(20,32,21,31)(22,33,24,35)(26,43,44,40)(27,45,37,41)(28,42,39,46)(36,48,38,47);(9,11,12,15)(10,14,13,16)(17,20,23,21)(18,26,29,44)(19,31,34,32)(22,36,24,38)(25,40,30,43)(27,28,37,39)(33,47,35,48)(41,42,45,46);(2,6)(5,7)(10,13)(14,16)(17,21,23,20)(18,27,29,37)(19,32,34,31)(22,36,24,38)(25,41,30,45)(26,28,44,39)(33,47,35,48)(40,42,43,46);(3,8)(5,7)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,36,21,38)(25,42,30,46)(26,37,44,27)(31,47,32,48)(40,45,43,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,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, 6664523);