#146579

order	512
nilpotency class	4
isoclinism	isoclinic class of #146575
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10735
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1575 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1575 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53775);