#507717

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 414913);