#503772

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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=f,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 410968);