#507090

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

410 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)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,28,35)(20,24,33,40)(21,25,32,39)(27,34,41,45)(29,36,42,46)(30,37,43,47)(31,38,44,48);(1,3,5,4)(2,6,8,7)(17,19,20,29,32,43,44,41,22,28,33,42,21,30,31,27)(18,23,24,36,39,47,48,45,26,35,40,46,25,37,38,34);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,32)(23,34)(24,38)(25,39)(28,41)(29,30)(33,44)(35,45)(36,37)(40,48)(42,43)(46,47);(1,4,5,3)(2,7,8,6)(19,28)(23,35)(27,41)(29,42)(30,43)(34,45)(36,46)(37,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=ef,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 414286);