#72251923

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3943226
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#454002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3709 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=fghj,d^2=fgi,g^2=j,h^2=j,b^a=bi,c^a=cf,c^b=cfj,d^b=dfghi,d^c=dhi,e^b=egij,e^d=eg,f^b=fj,g^c=gj,g^e=gj,h^b=hj,h^c=hj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251959
maximal normal subgroup of	#72251931, #72251959

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