#6631469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1689 conjugacy classes of subgroups — too many to draw. Only its 488 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538665);