#72335360

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 10810 conjugacy classes of subgroups; only the 3345 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2)(3,4)(7,11)(9,12)(13,14,17,19)(15,16,20,18)(23,27)(25,28);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,18,19,16)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(7,11)(9,12)(14,19)(16,18)(21,22)(23,28)(24,26)(25,27);(2,4)(7,11)(9,12);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,16,17,18)(14,20,19,15)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^10|a^2=g,b^2=gj,e^2=gh,f^2=i,b^a=bg,c^a=cg,c^b=cgi,d^a=dgh,d^b=dgh,d^c=dgh,e^a=eh,e^b=ehj,e^c=egh,e^d=egh,f^a=fj,f^c=fgj,f^d=fhij,f^e=fhj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335444
maximal normal subgroup of	#72335444
maximal quotient of	#72336078

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