#72249663

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32899
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2308 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=i,d^2=ei,g^2=j,b^a=bi,d^a=df,d^b=dg,d^c=de,e^b=eh,e^c=ej,e^d=ej,f^b=fi,g^a=gi,g^c=gh,g^d=ghj,g^f=gj,i^d=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255636, #72255775, #72255854, …
maximal normal subgroup of	#72255636, #72255775, #72255854, …
maximal quotient of	#72261677, #72263138, #72263325

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