#500737

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24236
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407933);