#406531

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 313727);