#493628

order	512
nilpotency class	4
isoclinism	isoclinic class of #493621
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×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62886	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2308 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fi,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, 400824);