#72231950

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

27461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=i,c^2=f,d^2=i,e^2=ij,f^2=i,g^2=i,h^2=i,c^a=cf,c^b=ci,d^c=dg,e^a=ej,e^c=eh,e^d=ej,f^a=fi,f^e=fj,g^c=gi,g^d=gi,g^e=gij,h^a=hj,h^c=hij,h^d=hij,h^e=hi

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232732, #72234695, #72234791, …
maximal normal subgroup of	#72232732, #72234695, #72234791, …
maximal quotient of	#72231951, #72251772, #72258050, …

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