#495680

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₈×#120
prime radical	#3018	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

922 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224961, C₂×#495680, #72341729, …
maximal normal subgroup of	#72224961, C₂×#495680, #72341733, …
maximal quotient of	C₂×#495680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402876);