#499619

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 406815);