#128068

order	512
nilpotency class	4
isoclinism	isoclinic class of #128067
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8101
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

1249 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1249 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,h^a=hi,d^b=dh,e^b=eg,g^b=gi,d^c=di,f^e=fi

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, 35264);