#72319947

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3860027
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5603 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=gj,h^2=j,b^a=bgj,c^a=cg,c^b=cgj,d^a=dh,d^b=dh,d^c=df,e^a=eghij,e^b=ei,e^c=egij,f^c=fj,g^d=gj,h^b=hj,h^c=hj,h^d=hj,h^e=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320148, #72338568
maximal normal subgroup of	#72320148, #72338568

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