#133755

order	512
nilpotency class	4
isoclinism	isoclinic class of #133293
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×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 640 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,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, 40951);