#72334424

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

843482 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^11|a^2=g,b^2=ghi,b^a=bgh,c^a=cj,c^b=cjk,d^a=dijk,d^b=dij,d^c=dk,e^a=ehi,e^b=ehi,f^a=fg,f^b=fghi,f^c=fj,f^d=fij,f^e=fhi

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334430, #72339327
maximal normal subgroup of	#72334430, #72339327

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