#496286

order	512
nilpotency class	4
isoclinism	isoclinic class of #496285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21758
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60462	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1004 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)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,19)(11,26)(12,25)(14,41,43,50)(15,45,23,57)(17,32,47,40)(20,56)(21,52)(24,39)(27,51)(29,38)(30,42,33,53)(31,48,37,59)(34,55)(35,60)(36,54)(44,62,49,64)(46,61,58,63);(1,3)(2,8)(4,14)(5,23)(6,17)(7,16)(9,33)(10,13)(11,37)(12,32)(15,27)(18,50)(19,53)(20,44)(21,43)(22,45)(24,58)(25,47)(26,59)(28,40)(29,49)(30,34)(31,36)(35,63)(38,62)(39,61)(41,52)(42,55)(46,60)(48,54)(51,57)(56,64);(1,4,7,21)(2,9,10,34)(3,14,16,43)(5,24,27,60)(6,29,25,20)(8,30,13,33)(11,38,36,56)(12,39,28,35)(15,46,23,58)(17,49,47,44)(18,51,52,22)(19,54,55,26)(31,62,37,64)(32,63,40,61)(41,57,50,45)(42,59,53,48);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,30)(10,13)(11,31)(12,32)(18,50)(19,42)(20,44)(21,43)(22,57)(23,27)(24,46)(25,47)(26,48)(28,40)(29,49)(33,34)(35,61)(36,37)(38,62)(39,63)(41,52)(45,51)(53,55)(54,59)(56,64)(58,60)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403482);