#493626

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	#62871	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2307 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 400822);