#507207

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

468 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=ei,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414403);