#72268342

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	D₄×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5898 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:

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

Polycyclic presentation:

2^10|a^2=ij,c^2=ij,d^2=j,h^2=j,b^a=bh,c^b=cj,d^b=dg,d^c=dg,e^a=eij,e^b=ef,e^c=efi,e^d=eij,f^d=fi,g^e=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268377, #72268455, #72310658
maximal normal subgroup of	#72268377, #72268455, #72310658

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