#72256786

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 9099 conjugacy classes of subgroups; only the 3441 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=ghij,d^2=j,e^2=j,b^a=bh,c^a=ci,c^b=cg,d^a=dj,d^b=di,d^c=dhj,e^c=ei,e^d=ej,f^c=fhj,f^d=fj

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256795
maximal normal subgroup of	#72256795

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