#499753

order	512
nilpotency class	4
isoclinism	isoclinic class of #499752
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 ◁ #2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1246 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)(3,16)(4,21)(5,26)(6,15)(7,13)(8,14)(9,18)(10,19)(11,22)(12,31)(17,34)(20,45)(23,43)(24,54)(25,55)(27,46)(28,36)(29,40)(30,37)(32,44)(33,39)(35,59)(38,60)(41,58)(42,57)(47,48)(49,53)(50,63)(51,61)(52,62)(56,64);(1,3)(2,9)(4,17)(5,27)(6,19)(7,33)(8,20)(10,37)(11,34)(12,39)(13,35)(14,36)(15,46)(16,26)(18,31)(21,47)(22,51)(23,49)(24,52)(25,50)(28,30)(29,48)(32,59)(38,54)(40,62)(41,63)(42,53)(43,61)(44,45)(55,64)(56,57)(58,60);(1,3)(2,10)(4,17)(5,28)(6,33)(7,19)(8,20)(9,12)(11,38)(13,45)(14,39)(15,16)(18,32)(21,53)(22,56)(23,52)(24,49)(25,50)(26,35)(27,30)(29,60)(31,59)(34,43)(36,37)(40,47)(41,61)(42,64)(44,46)(48,58)(51,57)(54,63)(55,62);(1,4,8,25)(2,11,14,41)(3,17,20,50)(5,29,32,57)(6,24,7,23)(9,34,36,63)(10,38,39,61)(12,43,37,54)(13,42,15,40)(16,47,45,64)(18,51,28,60)(19,52,33,49)(21,44,55,26)(22,30,58,31)(27,48,59,56)(35,53,46,62)

Polycyclic presentation:

2^9|d^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,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307105, #72307114, #72307115, …
maximal normal subgroup of	#72307105, #72307114, #72307115, …
maximal quotient of	C₂×#499753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406949);