#133729

order	512
nilpotency class	4
isoclinism	isoclinic class of #133281
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, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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^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, 40925);