#132173

order	512
nilpotency class	4
isoclinism	isoclinic class of #131997
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 618 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fi,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, 39369);