#509974

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

496 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)(3,5)(4,6)(7,8)(9,10,13,19,34,49,61,44,15,21,35,43,14,20,32,17)(11,16,24,38,25,39,58,65,26,41,56,36,22,40,51,42)(12,27,28,45,29,59,62,47,31,60,63,46,30,48,33,18)(23,52,53,69,50,64,70,67,55,71,72,66,54,68,57,37);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,23)(13,24)(14,22)(15,26)(17,42)(18,37)(19,38)(20,40)(21,41)(25,34)(27,52)(28,53)(29,50)(30,54)(31,55)(32,51)(33,57)(35,56)(36,43)(39,49)(44,65)(45,69)(46,66)(47,67)(48,68)(58,61)(59,64)(60,71)(62,70)(63,72);(1,4,5,8)(2,6,3,7)(10,17)(11,22)(13,32)(14,34)(16,36)(18,27)(19,20)(21,44)(23,50)(24,56)(25,26)(28,33)(29,30)(35,61)(37,64)(38,41)(39,65)(40,42)(43,49)(45,48)(46,59)(47,60)(52,69)(54,55)(57,70)(62,63)(66,71)(67,68);(2,3)(6,7)(9,12,15,31)(10,18,21,47)(11,23,26,55)(13,33,35,62)(14,29,34,30)(16,37,41,67)(17,27,44,60)(19,48,43,59)(20,45,49,46)(22,50,25,54)(24,57,56,70)(28,61,63,32)(36,64,38,68)(39,66,40,69)(42,52,65,71)(51,53,58,72)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 417170);