#356057

order	512
nilpotency class	3
isoclinism	isoclinic class of #356055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2187 conjugacy classes of subgroups; only the 472 normal subgroups are shown.

normal subgroups only

This group has 2187 conjugacy classes of subgroups — too many to draw. Only its 472 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263253);