#72338635

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#46428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#1082861
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gi,d^2=gj,e^2=ij,f^2=j,g^2=j,b^a=bf,c^a=ch,c^b=chi,d^a=dh,d^b=di,d^c=dh,e^a=eghij,e^b=eg,e^d=egij,f^e=fj,g^b=gj,h^b=hj,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338637
maximal normal subgroup of	#72338637

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.7);;
gap> SetPower(coll, 2, F.7*F.9);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);