#72237405

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 46462 conjugacy classes of subgroups; only the 444288 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 20 points:

(5,6)(7,12)(8,10)(9,11);(5,7,9,10)(6,8,11,12);(5,6)(7,12)(8,10)(9,11)(13,14)(15,18)(16,19)(17,20);(5,7,9,10)(6,8,11,12)(13,15)(14,17)(16,20)(18,19);(5,6)(7,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19);(2,4)(13,15)(14,18)(16,20)(17,19);(1,2)(3,4)(13,14)(15,17)(16,19)(18,20);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)

Polycyclic presentation:

2^11|b^2=j,d^2=j,b^a=bj,c^b=cj,d^a=dj,d^c=djk,g^e=gk,g^f=gi,h^e=hj,h^f=hik

Conjugacy classes & character table

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