#133237

order	512
nilpotency class	4
isoclinism	isoclinic class of #133089
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 597 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)(3,11,14,23)(4,15,17,8)(5,9)(6,10)(7,13,21,18)(12,26)(16,30,32,22)(19,24)(20,27)(25,28)(29,31)(33,34,37,42,38,41,51,56)(35,43,46,57,45,39,52,54)(36,47,49,58,48,40,50,55)(44,59,61,64,60,53,62,63);(1,3,6,14)(2,7,10,21)(4,12,17,28)(5,18,19,13)(8,20,15,29)(9,23,24,11)(16,31,32,27)(22,26,30,25)(33,35)(34,39)(36,44)(37,46)(38,45)(40,53)(41,43)(42,57)(47,59)(48,60)(49,61)(50,62)(51,52)(54,56)(55,64)(58,63);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,25)(13,27)(14,28)(18,31)(19,32)(21,29)(23,26)(24,30)(33,36,38,48)(34,40,41,47)(35,44,45,60)(37,50,51,49)(39,53,43,59)(42,58,56,55)(46,62,52,61)(54,64,57,63)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fhi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40433);