#72328294

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 10118 conjugacy classes of subgroups; only the 3309 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)(6,9)(10,11)(14,17)(15,19)(21,22,25,28)(23,26,24,27);(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,16,17)(15,20,19,18)(23,24)(26,27);(1,2)(3,4)(6,9)(10,11)(15,19)(18,20)(21,23,25,24)(22,26,28,27);(1,2,3,4)(6,9)(7,12)(14,17)(15,19)(21,24,25,23)(22,27,28,26);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28)

Polycyclic presentation:

2^10|a^2=i,b^2=h,c^2=hj,d^2=i,f^2=ghj,b^a=bij,c^a=chij,c^b=cg,d^b=dhi,d^c=dhij,e^a=eh,e^b=eh,e^c=eh,e^d=eh,f^a=fgij,f^b=fh,f^c=fhj,f^d=fhi,f^e=fghj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328397
maximal normal subgroup of	#72328397

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