#504010

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×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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)(3,7)(4,5)(6,8)(9,10)(11,25)(12,32)(13,22)(14,21)(15,23)(16,41)(17,24)(18,28)(19,29)(20,35)(26,42)(27,48)(30,49)(31,55)(33,50)(34,47)(36,63)(37,51)(38,69)(39,64)(40,44)(43,68)(45,58)(46,56)(52,65)(53,57)(54,59)(60,71)(61,70)(62,67)(66,72);(2,5)(6,7)(9,11,14,29,16,30,26,56,17,31,34,58,15,40,13,28)(10,18,22,44,23,45,47,55,24,46,42,49,41,19,21,25)(12,27,36,60,38,61,57,66,39,33,65,67,37,68,35,59)(20,43,51,62,52,50,64,72,53,70,69,71,63,48,32,54);(9,11,26,56,15,40,14,29,17,31,13,28,16,30,34,58)(10,19,47,44,41,46,22,25,24,45,21,49,23,18,42,55)(12,33,57,59,37,61,36,67,39,27,35,66,38,68,65,60)(20,50,64,71,63,43,51,72,53,48,32,62,52,70,69,54);(1,3,4,8)(2,6,5,7)(9,12,34,65,16,38,13,35,17,39,14,36,15,37,26,57)(10,20,42,69,23,52,21,32,24,53,22,51,41,63,47,64)(11,27,58,67,30,61,28,59,31,33,29,60,40,68,56,66)(18,43,49,71,45,50,25,54,46,70,44,62,19,48,55,72)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=ehi,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, 411206);