#133747

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,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, 40943);