#72338951

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#20609
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	#8934366	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

5126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7,10,25)(3,13)(5,18)(8,21,12,24)(9,27,11,22)(14,31)(15,29)(16,32)(19,30,26,23)(20,28);(1,2,3,7,4,10,13,25)(5,19,20,22,16,26,15,27)(6,8,14,21,17,12,31,24)(9,28,23,18,11,29,30,32);(2,8)(5,16)(7,21)(9,19)(10,12)(11,26)(15,20)(18,32)(22,23)(24,25)(27,30)(28,29);(2,9,10,11)(3,14)(7,22,25,27)(8,26,12,19)(13,31)(15,28)(20,29)(21,30,24,23)

Polycyclic presentation:

2^10|a^2=hj,b^2=ef,d^2=hj,g^2=j,b^a=be,c^b=chij,d^a=dhj,d^b=dg,d^c=dhj,e^a=ej,e^b=eh,e^d=ei,f^a=fhj,f^b=fh,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^e=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338955, #72338965, #72338969, …
maximal normal subgroup of	#72338955, #72338965, #72338969, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8*F.10);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);