#132164

order	512
nilpotency class	4
isoclinism	isoclinic class of #131988
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 621 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) = 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)(35,43)(36,40)(37,50,52,41)(38,42)(39,45,54,64)(44,61)(46,60)(47,59)(48,58,51,56)(49,57)(53,62,55,63);(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)(34,39)(36,47)(37,45)(38,46)(40,55)(41,43)(42,54)(44,49)(48,63)(50,60)(51,62)(52,64)(53,57)(56,59)(58,61);(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)(33,36,38,49)(34,40,42,57)(35,44,46,47)(37,51,52,48)(39,53,54,55)(41,58,50,56)(43,59,60,61)(45,63,64,62)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,f^2=h,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, 39360);