#505445

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

432 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)(12,14)(13,16)(17,18,21,25)(19,28,31,24)(20,33,23,26)(22,27,35,42)(29,38,48,44)(30,39,37,41)(32,46,47,40)(34,45,36,43);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,37)(25,28)(26,39)(27,44)(32,36)(33,41)(34,47)(35,48)(38,42)(40,43)(45,46);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,16,14)(18,25)(19,29)(20,23)(22,36)(24,38)(27,45)(28,44)(30,32)(31,48)(34,35)(37,47)(39,40)(41,46)(42,43);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,20,21,23)(18,26,25,33)(19,30,31,37)(22,34,35,36)(24,39,28,41)(27,43,42,45)(29,47,48,32)(38,46,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ei,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,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412641);