#493507

order	512
nilpotency class	4
isoclinism	isoclinic class of #493500
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3499 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2143 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)(3,15)(4,21,24,44)(5,11)(6,27,29,12)(7,13)(8,14)(9,18,36,50)(10,23,39,30)(16,40,52,34)(17,47)(19,48)(20,49)(22,43)(25,58)(26,42)(28,37,59,41)(31,38)(32,45)(33,53)(35,54,63,64)(46,51,62,55)(56,60)(57,61);(1,3,8,20)(2,9,14,36)(4,16,24,52)(5,25,22,17)(6,18,29,50)(7,19,32,57)(10,33,39,60)(11,40,43,34)(12,15,27,49)(13,35,45,63)(21,58,44,47)(23,51,30,55)(26,56,31,53)(28,54,59,64)(37,61,41,48)(38,46,42,62);(1,4,8,24)(2,10,14,39)(3,16,20,52)(5,6,22,29)(7,30,32,23)(9,33,36,60)(11,41,43,37)(12,38,27,42)(13,44,45,21)(15,46,49,62)(17,50,25,18)(19,55,57,51)(26,59,31,28)(34,61,40,48)(35,47,63,58)(53,54,56,64);(1,5,8,22)(2,11,14,43)(3,17,20,25)(4,6,24,29)(7,31,32,26)(9,34,36,40)(10,37,39,41)(12,21,27,44)(13,38,45,42)(15,47,49,58)(16,50,52,18)(19,56,57,53)(23,59,30,28)(33,48,60,61)(35,62,63,46)(51,54,55,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=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, 400703);