#494950

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,21,15,20,34,45,16,19,35,44,13,31,33,57)(11,17,24,39,25,38,53,64,26,37,54,63,23,50,52,69)(12,27,29,55,28,18,40,42,30,43,56,66,32,41,58,65)(22,46,48,67,47,36,59,61,49,62,68,72,51,60,70,71);(2,5)(6,8)(9,11,14,24,15,25,34,53,16,26,35,54,13,23,33,52)(10,17,21,39,20,38,45,64,19,37,44,63,31,50,57,69)(12,22,29,48,28,47,40,59,30,49,56,68,32,51,58,70)(18,36,42,61,43,62,66,72,41,60,65,71,27,46,55,67);(9,12)(10,18)(11,22)(13,32)(14,29)(15,28)(16,30)(17,36)(19,41)(20,43)(21,42)(23,51)(24,48)(25,47)(26,49)(27,31)(33,58)(34,40)(35,56)(37,60)(38,62)(39,61)(44,65)(45,66)(46,50)(52,70)(53,59)(54,68)(55,57)(63,71)(64,72)(67,69);(1,3,4,7)(2,6,5,8)(10,19)(17,37)(18,41)(20,31)(21,44)(27,43)(36,60)(38,50)(39,63)(42,65)(45,57)(46,62)(55,66)(61,71)(64,69)(67,72)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402146);