#511548

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 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)(3,6,4,8)(10,13)(11,15)(17,18,22,26)(19,23,30,38)(20,24,33,41)(21,25,32,40)(27,35,44,47)(28,36,43,46)(29,37,34,42)(31,39,45,48);(1,2)(3,6)(4,8)(5,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,27)(21,34)(22,30)(24,35)(25,42)(26,38)(28,31)(29,32)(33,44)(36,39)(37,40)(41,47)(43,45)(46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(19,27)(20,31)(21,32)(23,35)(24,39)(25,40)(28,29)(30,44)(33,45)(34,43)(36,37)(38,47)(41,48)(42,46);(1,4,5,3)(2,6,7,8)(10,13)(14,16)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 418744);