#357287

order	512
nilpotency class	3
isoclinism	isoclinic class of #357285
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13827
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

1495 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1495 conjugacy classes of subgroups — too many to draw. Only its 312 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264483);