#72337007

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 10054 conjugacy classes of subgroups; only the 3289 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)(5,6,9,8)(7,10,12,11)(13,14,17,15)(16,19,20,18)(21,22,26,27)(23,25,28,24);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,15)(18,19)(21,23,26,28)(22,24,27,25);(1,2)(3,4)(6,8)(10,11)(13,14,17,15)(16,19,20,18)(22,27)(24,25);(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,14,17,15)(16,19,20,18)(21,24,26,25)(22,23,27,28);(1,2,3,4)(5,8,9,6)(7,10,12,11)(13,15,17,14)(16,19,20,18)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,17,20)(14,18,15,19)(21,25,26,24)(22,28,27,23)

Polycyclic presentation:

2^10|a^2=i,b^2=ij,c^2=gij,d^2=hi,e^2=hj,f^2=g,b^a=bg,c^a=cgj,c^b=cgij,d^a=dh,d^b=dhj,d^c=dghj,e^a=ehij,e^b=ehi,e^c=eghi,e^d=eij,f^a=fj,f^c=fg,f^d=fghij,f^e=fhij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

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