#72233423

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#10586472	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

22657 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^a=cf,c^b=cfij,d^a=dgij,d^b=dgj,e^a=eh,e^b=ehi,e^c=ej,f^a=fi,f^b=fi,f^d=fi,f^e=fj,g^c=gi,h^c=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233437, #72233450, #72233486, …
maximal normal subgroup of	#72233437, #72233450, #72233486, …

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