#508591

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=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, 415787);