#72328211

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 10544 conjugacy classes of subgroups; only the 3625 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=ij,e^2=gh,f^2=hij,b^a=bi,c^a=cij,c^b=ch,d^b=dij,d^c=dij,e^a=eij,e^b=eg,e^c=eghij,e^d=eij,f^a=fij,f^b=fhij,f^d=fhij,f^e=fij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328270
maximal normal subgroup of	#72328270
maximal quotient of	#72331521

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