#72337355

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 12255 conjugacy classes of subgroups; only the 16160 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:

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

Polycyclic presentation:

2^11|a^2=g,d^2=ghjk,e^2=gij,b^a=bh,c^a=cj,d^a=di,d^b=dijk,d^c=djk,e^a=ehk,e^b=eijk,e^c=ej,e^d=ehik,f^a=fj,f^d=fj,f^e=fj

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337359, #72340056
maximal normal subgroup of	#72337359, #72340056

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