#504716

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

511 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)(34,52)(35,50)(37,57)(40,54)(41,69)(42,53)(43,70)(47,66)(55,58)(56,59)(60,67)(61,72)(62,64)(63,68)(65,71);(1,3,5,4)(2,6,7,8)(9,11,15,30,17,31,14,29,18,12,33,36,16,37,27,35)(10,19,23,48,24,49,22,26,25,50,44,57,46,20,45,32)(13,28,40,60,42,61,39,59,43,34,63,65,41,66,58,64)(21,47,53,71,54,52,38,56,55,72,69,67,68,51,70,62);(1,4,5,3)(2,6,7,8)(9,12,14,35,16,31,15,36,18,11,27,29,17,37,33,30)(10,20,22,48,46,50,23,32,25,49,45,57,24,19,44,26)(13,28,39,59,41,66,40,60,43,34,58,64,42,61,63,65)(21,52,38,67,68,47,53,56,55,51,70,71,54,72,69,62);(2,7)(6,8)(9,13,15,40,17,42,14,39,18,43,33,63,16,41,27,58)(10,21,23,53,24,54,22,38,25,55,44,69,46,68,45,70)(11,28,30,60,31,61,29,59,12,34,36,65,37,66,35,64)(19,47,48,71,49,52,26,56,50,72,57,67,20,51,32,62)

Polycyclic presentation:

2^9|b^2=ehi,c^2=e,d^2=f,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,d^c=di,g^c=gi

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, 411912);