#72339939

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 10544 conjugacy classes of subgroups; only the 8752 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=g,c^2=k,f^2=gk,c^a=cj,d^a=di,d^c=dk,e^a=ehijk,f^a=fghk,f^b=fg,f^c=fj,f^d=fi,f^e=fhijk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339945
maximal normal subgroup of	#72339945

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