#504933

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,41)(28,36)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,45)(42,47)(46,48);(2,6)(7,8)(10,13)(14,15)(17,19,20,28)(18,23,24,36)(21,27,31,45)(22,34,32,30)(25,41,39,35)(26,44,40,38)(29,47,46,33)(37,42,48,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(18,24)(19,27)(21,33)(22,32)(23,35)(25,42)(28,45)(29,30)(31,47)(34,46)(36,41)(37,44)(38,48)(39,43);(1,4,5,3)(2,7,6,8)(10,13)(14,15)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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, 412129);