#72340045

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	C₂⁶×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

836048 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)(14,17)(18,19);(13,14,16,17)(15,18,20,19);(5,6)(7,8)(21,22,25,23)(24,26,28,27);(2,4)(6,8)(9,10,11,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(13,15)(14,18)(16,20)(17,19)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^11|b^2=i,c^2=k,d^2=g,b^a=bi,d^a=dh,d^b=di,d^c=dj,e^d=ehi,f^d=fik

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340056
maximal normal subgroup of	#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, 2, F.9);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);