#72339043

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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=j,e^2=i,b^a=bj,c^a=cf,c^b=cgh,d^a=dj,d^b=dfi,e^a=eg,e^b=egi,e^c=egh,e^d=eghi,f^b=fj,f^e=fi,g^c=gij,g^d=gj,h^a=hj,h^b=hij,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339048, #72340222, #72340248, …
maximal normal subgroup of	#72339048, #72340222, #72340248, …

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