#416707

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

1122 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=ef,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 323903);