#133795

order	512
nilpotency class	4
isoclinism	isoclinic class of #133474
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 468 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,e^b=eh,f^b=fg,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, 40991);