#512223

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419419);