#72251920

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3554137
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#394321	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3413 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)(3,11,13,26)(4,15,17,9)(5,10)(6,7)(8,14,20,29)(12,25)(16,24,18,23)(19,21)(22,28)(27,32)(30,31);(2,7)(10,21)(12,27)(28,30);(1,3)(2,8,7,20)(4,12)(5,14)(6,13)(9,22,15,31)(10,11,21,26)(16,28)(17,27)(18,30)(19,29)(23,32,24,25);(1,4,5,18,6,17,19,16)(2,9,21,23,7,15,10,24)(3,12,14,30,13,27,29,28)(8,22,26,32,20,31,11,25);(4,16)(5,19)(9,23)(10,21)(11,26)(12,28)(14,29)(15,24)(17,18)(22,32)(25,31)(27,30)

Polycyclic presentation:

2^10|a^2=g,c^2=fgi,d^2=fh,h^2=j,b^a=bgj,c^a=ch,c^b=cf,d^a=dfij,d^b=dfg,e^a=ei,e^d=eghi,f^a=fj,f^d=fj,g^c=gj,g^d=gj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251940, #72251956
maximal normal subgroup of	#72251931, #72251940, #72251956

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