#72240462

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	#2032655
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#453952	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4649 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)(3,7)(4,8)(5,17,18,9)(6,10)(11,22)(12,28,19,23)(13,21)(14,24)(15,30)(16,25,20,26)(27,31,29,32);(4,14)(5,18)(7,21)(8,24)(9,17)(11,15)(23,28)(27,29);(1,3)(2,7)(4,15)(5,19)(6,13)(8,22)(9,23)(10,21)(11,14)(12,18)(16,29)(17,28)(20,27)(24,30)(25,32)(26,31);(1,4)(2,8)(3,11,13,15)(5,20)(6,14)(7,22,21,30)(9,26)(10,24)(12,29,19,27)(16,18)(17,25)(23,32,28,31);(1,5)(2,9,10,17)(3,12,13,19)(4,16,14,20)(6,18)(7,23)(8,25)(11,27)(15,29)(21,28)(22,31,30,32)(24,26)

Polycyclic presentation:

2^10|a^2=g,d^2=i,e^2=hi,b^a=bi,c^a=cf,c^b=cfh,d^a=dg,d^b=dj,d^c=dgh,e^a=eghj,e^b=eij,e^c=egh,e^d=eg,f^d=fj,f^e=fj,g^e=gj,h^a=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240477, #72251093, #72251291, …
maximal normal subgroup of	#72240477, #72251093, #72251291, …

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