#504713

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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

513 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)(3,8)(4,6)(5,7)(9,10)(11,26)(12,32)(13,38)(14,44)(15,45)(16,24)(17,46)(18,25)(19,29)(20,36)(21,39)(22,33)(23,27)(28,51)(30,49)(31,48)(34,63)(35,50)(37,56)(40,53)(41,69)(42,52)(43,70)(47,66)(54,57)(55,58)(59,72)(60,71)(61,64)(62,68)(65,67);(1,3,5,4)(2,6,7,8)(9,11,15,30,17,31,14,29,18,12,33,36,16,37,27,35)(10,19,23,48,24,49,22,26,25,50,44,56,46,20,45,32)(13,28,40,59,42,60,39,58,43,34,62,65,41,66,57,64)(21,47,52,67,53,63,38,55,54,71,69,72,68,51,70,61);(1,4,5,3)(2,6,7,8)(9,12,14,35,16,31,15,36,18,11,27,29,17,37,33,30)(10,20,22,48,46,50,23,32,25,49,45,56,24,19,44,26)(13,34,39,64,41,60,40,65,43,28,57,58,42,66,62,59)(21,51,38,67,68,71,52,61,54,63,70,72,53,47,69,55);(2,7)(6,8)(9,13,33,62,17,42,27,57,18,43,15,40,16,41,14,39)(10,21,44,69,24,53,45,70,25,54,23,52,46,68,22,38)(11,28,36,65,31,60,35,64,12,34,30,59,37,66,29,58)(19,47,56,72,49,63,32,61,50,71,48,67,20,51,26,55)

Polycyclic presentation:

2^9|b^2=ehi,c^2=e,d^2=fi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411909);