#392241

order	512
nilpotency class	3
isoclinism	isoclinic class of #392239
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14681
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3279 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 3279 conjugacy classes of subgroups — too many to draw. Only its 432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299437);