#132071

order	512
nilpotency class	4
isoclinism	isoclinic class of #132067
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8456
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#62276	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

553 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39267);