#72285239

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	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,d^2=gh,e^2=gh,b^a=bi,c^a=cf,c^b=cf,d^a=dfi,d^c=dfg,e^a=efhij,e^b=eghj,e^c=efij,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^d=gj,g^e=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285772, #72340248, #72340261, …
maximal normal subgroup of	#72285772, #72340248, #72340261, …

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