#508820

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

430 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18,20,24)(19,27,29,23)(21,25,31,38)(22,26,32,39)(28,43,45,36)(30,42,35,37)(33,40,44,48)(34,47,46,41);(9,11,13,12)(10,14,15,16)(19,28)(21,33)(22,32)(23,36)(25,40)(26,39)(27,43)(29,45)(30,46)(31,44)(34,35)(37,47)(38,48)(41,42);(2,5)(6,8)(11,12)(14,16)(17,19,20,29)(18,23,24,27)(21,34,31,46)(22,35,32,30)(25,41,38,47)(26,42,39,37)(28,44,45,33)(36,48,43,40);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,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, 416016);