#6432153

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432152
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2967 conjugacy classes of subgroups; only the 904 normal subgroups are shown.

normal subgroups only

This group has 2967 conjugacy classes of subgroups — too many to draw. Only its 904 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339349);