#72310686

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#10577998	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5817 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=h,d^2=fij,e^2=ij,f^2=j,c^a=ch,c^b=cij,d^a=df,d^c=dg,e^a=efghi,e^c=ei,e^d=efghi,f^a=fj,f^c=fi,f^e=fj,g^c=gi,g^d=gi,h^d=hi

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310707, #72311611
maximal normal subgroup of	#72310707, #72311611

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