#72241128

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

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

At least 13464 conjugacy classes of subgroups; only the 29513 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|d^2=i,f^2=h,g^2=j,b^a=bi,c^a=cj,c^b=ci,d^a=dj,d^b=di,d^c=di,f^a=fj,f^b=fi,f^c=fh,f^d=fhi,g^a=ghij,g^b=ghi,g^c=gj,g^e=ghi

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241176, #72241455
maximal normal subgroup of	#72241176, #72241455
maximal quotient of	#72328746, #72329012, #72329242, …

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