#6431949

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#394
prime radical	#63090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339145);