#6750063

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6657259);