#505103

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

476 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,23)(19,27,29,24)(21,25,31,36)(22,26,32,37)(28,38,46,39)(30,44,35,40)(33,43,45,41)(34,48,47,42);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(18,23)(19,28)(21,33)(22,32)(24,38)(25,41)(27,39)(29,46)(30,47)(31,45)(34,35)(36,43)(40,42)(44,48);(10,13)(14,15)(17,19,20,29)(18,24,23,27)(21,34,31,47)(22,35,32,30)(25,42,36,48)(26,44,37,40)(28,45,46,33)(38,43,39,41);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(18,23)(19,29)(25,36)(26,37)(28,46)(30,35)(34,47)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=i,d^2=fi,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, 412299);