#72331500

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂×#647656	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 18246 conjugacy classes of subgroups; only the 7504 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^11|e^2=h,f^2=i,b^a=bk,c^b=cg,e^a=ej,e^b=ei,e^d=ej,f^a=fj,f^b=fi,f^d=fi,f^e=fg

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331654
maximal normal subgroup of	#72331654

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