#72339936

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶×C₄

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

2660907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^11|a^2=h,c^2=jk,d^2=i,b^a=bh,c^a=chj,c^b=cg,d^a=dhi,d^b=dgij,d^c=dijk,e^a=ehijk,e^b=ehijk,e^c=ehijk,e^d=ehijk,f^a=fgjk,f^b=fgjk,f^c=fgjk,f^d=fgjk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339945, #72339997
maximal normal subgroup of	#72339945, #72339997

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