#72326574

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 10597 conjugacy classes of subgroups; only the 3321 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)(5,6)(7,9)(8,10)(11,12);(2,4)(14,15)(16,20)(21,22)(23,25)(24,26)(27,28);(1,2,3,4)(5,6,9,7)(8,11,12,10)(14,15)(18,19)(22,23)(26,27);(2,4)(6,7)(10,11)(13,14,17,15)(16,19,20,18)(21,22,25,23)(24,26,28,27);(2,4)(5,7,9,6)(8,10,12,11)(13,15,17,14)(16,19,20,18)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^10|c^2=j,d^2=hi,e^2=ghij,b^a=bg,c^a=cj,c^b=ch,d^a=dj,d^b=dhi,d^c=dhij,e^a=ej,e^b=ehi,e^c=eghi,e^d=egj,f^b=fi,f^c=fj,f^d=fi,f^e=fghj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326659
maximal normal subgroup of	#72326659

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