#493602

order	512
nilpotency class	4
isoclinism	isoclinic class of #493594
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1790
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62886	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2302 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=di,f^b=fhi,g^b=gh,e^c=eh,g^c=gi,f^e=fi,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, 400798);