#499749

order	512
nilpotency class	4
isoclinism	isoclinic class of #499746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1244 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)(3,17)(4,23)(5,26)(6,30)(7,16)(8,14)(9,15)(10,19)(11,25)(12,27)(13,34)(18,39)(20,47)(21,52)(22,53)(24,59)(28,51)(29,61)(31,40)(32,38)(33,50)(35,49)(36,45)(37,48)(41,57)(42,56)(43,63)(44,54)(46,62)(55,64)(58,60);(1,3,9,22)(2,10,15,41)(4,5,18,29)(6,31,35,56)(7,21,8,20)(11,43,45,38)(12,39,46,23)(13,47,50,52)(14,42,16,40)(17,49,53,30)(19,33,57,34)(24,32,54,63)(25,37,36,28)(26,44,61,59)(27,55,62,60)(48,64,51,58);(1,4,9,18)(2,11,15,45)(3,5,22,29)(6,24,35,54)(7,36,8,25)(10,38,41,43)(12,40,46,42)(13,44,50,59)(14,39,16,23)(17,51,53,48)(19,27,57,62)(20,37,21,28)(26,52,61,47)(30,58,49,64)(31,32,56,63)(33,55,34,60);(1,5)(2,12)(3,18)(4,22)(6,32)(7,28)(8,37)(9,29)(10,39)(11,40)(13,48)(14,43)(15,46)(16,38)(17,44)(19,55)(20,36)(21,25)(23,41)(24,56)(26,30)(27,34)(31,54)(33,62)(35,63)(42,45)(47,64)(49,61)(50,51)(52,58)(53,59)(57,60)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307503, #72307518, #72307547, …
maximal normal subgroup of	#72307503, #72307518, #72307547, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406945);