#6629957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,29,36)(20,24,32,40)(21,25,33,26)(28,34,41,46)(30,35,43,37)(31,38,44,39)(42,45,48,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,34)(25,35)(26,37)(27,36)(31,42)(32,41)(33,43)(38,45)(39,47)(40,46)(44,48);(10,14)(13,16)(17,20,22,32)(18,24,27,40)(19,28,29,41)(21,31,33,44)(23,34,36,46)(25,38,26,39)(30,42,43,48)(35,45,37,47);(18,25)(21,33)(23,35)(24,38)(26,27)(30,43)(31,44)(34,45)(36,37)(39,40)(42,48)(46,47);(1,3,4,7)(2,6,5,8)(19,29)(23,36)(28,41)(30,43)(34,46)(35,37)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537153);