#504791

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=efh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,e^d=eh,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, 411987);