#6624714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,23,27)(19,28,32,44)(20,33,35,48)(21,26,22,38)(24,30,40,31)(25,36,42,37)(29,41,34,39)(43,45,47,46);(2,6)(7,8)(10,14)(11,12)(17,19,22,31,23,32,21,30)(18,24,26,28,27,40,38,44)(20,34,37,45,35,29,36,46)(25,41,33,47,42,39,48,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,37,23,35,21,36)(18,25,26,33,27,42,38,48)(19,29,31,46,32,34,30,45)(24,39,28,43,40,41,44,47);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(20,35)(25,42)(29,34)(33,48)(36,37)(39,41)(43,47)(45,46);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,36,35,37)(24,28,40,44)(25,33,42,48)(29,45,34,46)(39,43,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=ehi

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, 6531910);