#507112

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

467 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,29)(28,36)(30,39)(31,40)(32,41)(33,42)(34,43)(35,38)(37,44)(45,47)(46,48);(1,3,4,8)(2,5,6,7)(17,19,28,44,32,35,34,33,20,29,21,30,22,31,46,45)(18,23,36,39,41,40,43,47,25,24,26,37,27,38,48,42);(10,13)(14,16)(17,19,20,29)(18,24,25,23)(21,33,28,45)(22,35,32,31)(26,42,36,47)(27,38,41,40)(30,46,44,34)(37,43,39,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 414308);