#500748

order	512
nilpotency class	4
isoclinism	isoclinic class of #500747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ 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)(3,15)(4,10)(5,12)(6,30)(7,33)(8,14)(9,19)(11,28)(13,27)(16,38)(17,50)(18,48)(20,54)(21,55)(22,45)(23,40)(24,60)(25,43)(26,46)(29,44)(31,37)(32,61)(34,35)(36,56)(39,53)(41,59)(42,58)(47,51)(49,64)(52,62)(57,63);(3,16,18,26)(5,6)(7,27)(9,35,37,54)(11,44)(12,30)(13,33)(15,46,48,38)(17,49,51,53)(19,20,31,34)(21,56,57,52)(22,32)(23,59)(24,58)(28,29)(36,55,62,63)(39,47,64,50)(40,41)(42,60)(45,61);(1,3)(2,9)(4,21)(5,26)(6,31)(7,34)(8,18)(10,39)(11,35)(12,15)(13,46)(14,37)(16,29)(17,24)(19,28)(20,27)(22,56)(23,53)(25,57)(30,54)(32,49)(33,38)(36,42)(40,50)(41,63)(43,64)(44,48)(45,55)(47,61)(51,58)(52,59)(60,62);(1,4,8,25)(2,10,14,43)(3,17)(5,22,29,59)(6,32,28,23)(7,24,27,58)(9,36)(11,40,30,61)(12,45,44,41)(13,42,33,60)(15,47)(16,49)(18,51)(19,52)(20,21)(26,53)(31,56)(34,57)(35,55)(37,62)(38,39)(46,64)(48,50)(54,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250066, #72307725
maximal normal subgroup of	#72250066, #72307725
maximal quotient of	#72250656, #72282220, #72282251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407944);