#72231727

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

31416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|c^2=i,d^2=i,e^2=fgj,g^2=j,c^a=ci,d^b=di,e^a=ef,e^b=eh,e^c=eg,e^d=efgj,f^b=fi,g^b=gi,g^c=gj,g^d=gj,h^a=hi,h^c=hi

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341937, #72342092
maximal subgroup of	#72268333, #72306071, C₂×#72231727, …
maximal normal subgroup of	#72268333, #72306071, C₂×#72231727, …
maximal quotient of	#72231730, #72250016, #72250237, …

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