#493662

order	512
nilpotency class	4
isoclinism	isoclinic class of #493653
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#63063	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2155 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400858);