#512250

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, 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

509 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,30,24,25)(20,26,21,35)(23,38,37,43)(27,34,45,36)(29,32,46,40)(31,44,33,41)(39,47,48,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,24)(18,25,28,30)(20,33,21,31)(23,32,37,40)(26,44,35,41)(27,42,45,47)(29,43,46,38)(34,48,36,39);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,21)(18,26,28,35)(19,31,24,33)(23,34,37,36)(25,41,30,44)(27,38,45,43)(29,42,46,47)(32,39,40,48);(17,21,22,20)(18,27,28,45)(19,31,24,33)(23,39,37,48)(25,42,30,47)(26,38,35,43)(29,41,46,44)(32,34,40,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,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,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, 419446);