#512228

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=f,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419424);