#499745

order	512
nilpotency class	4
isoclinism	isoclinic class of #499742
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 ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1267 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)(3,14)(4,10)(5,21)(6,24)(7,29)(8,13)(9,17)(11,22)(12,27)(15,44)(16,33)(18,43)(19,51)(20,41)(23,54)(25,35)(26,36)(28,53)(30,39)(31,40)(32,55)(34,50)(37,52)(38,56)(42,46)(45,60)(47,59)(48,57)(49,58)(61,64)(62,63);(1,3)(2,9)(4,15)(5,16)(6,25)(7,30)(8,18)(10,35)(11,33)(12,39)(13,34)(14,24)(17,27)(19,45)(20,50)(21,42)(22,49)(23,47)(26,46)(28,55)(29,32)(31,48)(36,60)(37,59)(38,57)(40,58)(41,44)(43,53)(51,62)(52,63)(54,61)(56,64);(1,4,8,7)(2,10,13,29)(3,15,18,30)(5,19,23,31)(6,20,28,27)(9,32,34,35)(11,36,37,38)(12,24,41,53)(14,39,43,44)(16,45,47,48)(17,25,50,55)(21,51,54,40)(22,26,52,56)(33,57,59,60)(42,58,61,62)(46,63,64,49);(1,5,8,23)(2,11,13,37)(3,16,18,47)(4,19,7,31)(6,26,28,56)(9,33,34,59)(10,36,29,38)(12,40,41,51)(14,42,43,61)(15,45,30,48)(17,49,50,63)(20,52,27,22)(21,53,54,24)(25,46,55,64)(32,57,35,60)(39,58,44,62)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307115, #72307122, #72307547, …
maximal normal subgroup of	#72307115, #72307122, #72307547, …
maximal quotient of	C₂×#499745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406941);