#510868

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,23,30,36)(20,25,33,39)(21,26,32,40)(27,35,44,46)(28,37,43,47)(29,38,34,42)(31,41,45,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,24,36)(20,27,33,44)(21,34,32,29)(25,35,39,46)(26,42,40,38)(28,45,43,31)(37,48,47,41);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(19,27)(20,31)(21,32)(23,35)(25,41)(26,40)(28,29)(30,44)(33,45)(34,43)(36,46)(37,38)(39,48)(42,47);(2,7)(3,8)(11,16)(13,15)(18,24)(23,36)(25,39)(26,40)(35,46)(37,47)(38,42)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418064);