#508485

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,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, 415681);