#132169

order	512
nilpotency class	4
isoclinism	isoclinic class of #131988
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,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, 39365);