#132122

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	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 467 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,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, 39318);