#507230

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	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

462 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)(28,41,44,35)(30,42,45,36)(31,43,46,37)(34,40,47,48);(1,3,5,4)(2,6,7,8)(17,19,21,30,33,46,47,44,20,29,32,45,22,31,34,28)(18,23,25,36,39,43,48,41,24,27,38,42,26,37,40,35);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(19,28)(21,34)(22,33)(23,35)(25,40)(26,39)(27,41)(29,44)(30,31)(32,47)(36,37)(38,48)(42,43)(45,46);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(19,29)(23,27)(28,44)(30,45)(31,46)(35,41)(36,42)(37,43)

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=ei,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 414426);