#132121

order	512
nilpotency class	4
isoclinism	isoclinic class of #132007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 74 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)(36,49,51,41)(37,42)(38,43)(39,46,55,45)(40,50,57,64)(47,56)(48,60)(52,58)(53,63)(54,62,59,61);(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)(33,35,38,48)(34,39,43,55)(36,46,51,45)(37,47,52,63)(40,56,57,53)(41,44,49,60)(42,54,58,59)(50,62,64,61);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,21)(10,26)(11,24)(12,25)(13,27)(15,31)(16,30)(17,29)(22,32)(23,28)(33,36)(34,40,43,57)(35,45)(37,50)(38,51)(39,53,55,56)(41,42,49,58)(44,59,60,54)(46,48)(47,61)(52,64)(62,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,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, 39317);