#72236189

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

64259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^10|c^2=hi,b^a=bh,c^a=chi,c^b=chi,d^a=dij,d^c=dij,e^a=ehi,e^c=ehi,e^d=ehj,f^a=fij,f^c=fj,g^b=ghi,g^c=ghi,g^d=gij,g^e=ghi

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72237738, #72240903, #72340816, …
maximal subgroup of	#72236721, #72236986, #72237738, …
maximal normal subgroup of	#72236721, #72236986, #72237738, …
maximal quotient of	#72297602, #72297949, C₂×#72236189
automorphism group of	#1902, #1903, #1904, …

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