#6624382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(3,8)(4,6)(5,7)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,26,22,27)(24,33,39,32)(25,35,42,36)(30,44,45,38)(43,47,48,46);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,38)(26,40)(27,29)(28,39)(34,48)(35,46)(36,47)(37,45)(41,43)(42,44);(9,10)(11,12)(13,14)(15,16)(17,20,23,37)(18,25,28,42)(19,30,31,45)(21,35,22,36)(24,38,39,44)(26,34,27,41)(29,43,40,48)(32,46,33,47);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,46,45,47)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531578);