#133220

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	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

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

normal subgroups only

This group has 628 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fh,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, 40416);