#511617

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18,21,26)(19,29,32,24)(20,25,23,28)(22,27,34,41)(30,45,46,38)(31,44,37,39)(33,40,35,42)(36,48,47,43);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,15,14)(19,30)(20,23)(22,35)(24,38)(25,28)(27,42)(29,45)(31,47)(32,46)(33,34)(36,37)(39,48)(40,41)(43,44);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,32)(18,24,26,29)(20,31,23,37)(22,36,34,47)(25,39,28,44)(27,43,41,48)(30,33,46,35)(38,40,45,42);(3,7)(5,8)(11,16)(13,14)(17,20,21,23)(18,25,26,28)(19,31,32,37)(22,33,34,35)(24,39,29,44)(27,40,41,42)(30,36,46,47)(38,43,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418813);