#6609157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(17,18,23,27)(19,24,30,41)(20,31,33,44)(21,26,22,37)(25,36,43,34)(28,45,32,48)(29,39,35,40)(38,46,42,47);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,32)(21,35)(22,29)(23,30)(25,42)(26,40)(27,41)(28,33)(31,48)(34,46)(36,47)(37,39)(38,43)(44,45);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,22,34,23,33,21,36)(18,25,26,31,27,43,37,44)(19,28,29,47,30,32,35,46)(24,38,40,45,41,42,39,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(20,33)(25,43)(28,32)(31,44)(34,36)(38,42)(45,48)(46,47);(10,13)(11,15)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,34,33,36)(24,39,41,40)(25,44,43,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6516353);