#493650

order	512
nilpotency class	4
isoclinism	isoclinic class of #493638
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×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63044	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2154 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,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, 400846);