#493403

order	512
nilpotency class	4
isoclinism	isoclinic class of #493397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3245 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,g^e=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, 400599);