#72256702

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 50987 conjugacy classes of subgroups; only the 444436 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 28 points:

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

Polycyclic presentation:

2^11|b^2=k,d^2=ij,e^2=j,b^a=bk,d^a=dj,d^b=di,d^c=di,e^a=ejk,f^d=fj,f^e=fjk,g^a=gk,g^b=gk,g^c=gk,g^e=gjk,g^f=gk,h^e=hk

Conjugacy classes & character table

362 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256706
maximal normal subgroup of	#72256706

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.11);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
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.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);