#72337285

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(9,10,13,16)(11,12,15,14)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4)(5,6,8,7)(11,15)(12,14)(17,19)(18,21)(20,24)(22,23);(1,2,3,4)(5,6)(7,8)(9,11,13,15)(10,14,16,12)(18,22)(21,23);(2,4)(6,7)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,16)(14,15)(17,19)(18,21)(20,24)(22,23);(9,12)(10,15)(11,16)(13,14)

Polycyclic presentation:

2^10|a^2=i,b^2=g,c^2=gh,e^2=g,b^a=bj,c^a=cij,c^b=ch,d^b=dhi,d^c=dhij,e^a=eij,e^c=eh,e^d=ehi,f^b=fi,f^c=fi,f^e=fi

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337290
maximal normal subgroup of	#72337290
maximal quotient of	#72337319, #72337354

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