#6432193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#397
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2709 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)(3,13,17,34)(4,9)(5,10)(6,11)(7,12)(8,16,28,24)(14,38,18,36)(15,37,41,57)(19,29)(20,30)(21,31)(22,32)(23,33)(25,35)(26,44,51,40)(27,42,52,49)(39,59,43,58)(45,53)(46,54)(47,55)(48,56)(50,61,63,60)(62,64);(1,3)(2,8)(4,18)(5,15)(6,24)(7,17)(9,26)(10,27)(11,34)(12,28)(13,35)(14,21)(16,25)(19,43)(20,40)(22,41)(23,49)(29,50)(30,38)(31,51)(32,52)(33,57)(36,55)(37,56)(39,45)(42,48)(44,47)(46,60)(53,63)(54,59)(58,64)(61,62);(1,4,6,20,7,21,25,47)(2,9,11,30,12,31,35,55)(3,14,16,40,17,18,24,44)(5,19,23,46,22,45,48,62)(8,26,13,36,28,51,34,38)(10,29,33,54,32,53,56,64)(15,39,42,60,41,43,49,61)(27,50,37,58,52,63,57,59);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,27)(9,29)(11,33)(12,32)(13,37)(14,39)(16,42)(17,41)(18,43)(20,46)(21,45)(24,49)(25,48)(26,50)(28,52)(30,54)(31,53)(34,57)(35,56)(36,58)(38,59)(40,60)(44,61)(47,62)(51,63)(55,64);(5,22)(10,32)(15,41)(19,45)(23,48)(27,52)(29,53)(33,56)(37,57)(39,43)(42,49)(46,62)(50,63)(54,64)(58,59)(60,61)

Polycyclic presentation:

2^9|a^2=g,c^2=f,f^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^c=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, 6339389);