#72236620

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 6098 conjugacy classes of subgroups; only the 445272 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^11|b^2=i,g^2=k,h^2=j,b^a=bi,g^a=gj,g^b=gjk,g^c=gj,g^d=gjk,g^e=gk,g^f=gi,h^e=hj

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251325, #72252166, #72252600, …
maximal normal subgroup of	#72251325, #72252166, #72252600, …
automorphism group of	#1977, #2002, #2003, …

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.9);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);