#494951

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 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)(9,10,14,20,15,19,34,44,16,21,35,45,13,31,33,57)(11,17,25,39,26,38,54,64,23,37,50,63,24,51,53,69)(12,27,29,55,28,18,40,41,30,42,56,65,32,43,58,66)(22,46,49,67,48,36,59,61,47,62,68,72,52,60,70,71);(2,5)(6,8)(9,11,14,25,15,26,34,54,16,23,35,50,13,24,33,53)(10,17,20,39,19,38,44,64,21,37,45,63,31,51,57,69)(12,22,29,49,28,48,40,59,30,47,56,68,32,52,58,70)(18,36,41,61,42,62,65,72,43,60,66,71,27,46,55,67);(9,12)(10,18)(11,22)(13,32)(14,29)(15,28)(16,30)(17,36)(19,42)(20,41)(21,43)(23,47)(24,52)(25,49)(26,48)(27,31)(33,58)(34,40)(35,56)(37,60)(38,62)(39,61)(44,65)(45,66)(46,51)(50,68)(53,70)(54,59)(55,57)(63,71)(64,72)(67,69);(1,3,4,7)(2,6,5,8)(11,23)(17,37)(22,47)(24,26)(25,50)(36,60)(38,51)(39,63)(46,62)(48,52)(49,68)(53,54)(59,70)(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,f^a=fi,d^b=di,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, 402147);