#72319297

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#442
commutator	C₂²×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#500990	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ #183301 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,13,10,7,11,15)(8,12,17,19,16,14,18,20)(22,28,23,31)(24,27)(25,32)(29,35)(30,34,33,36);(5,6,9,13,10,7,11,15)(8,12,17,19,16,14,18,20)(21,22,29,36,24,33,35,28,26,23,32,34,27,30,25,31);(5,7,9,15,10,6,11,13)(8,12,17,19,16,14,18,20)(21,23,27,30,26,22,24,33)(25,34,32,31,35,36,29,28);(2,4)(5,8,11,18,10,16,9,17)(6,12,15,20,7,14,13,19)(21,24,26,27)(22,23)(25,32,35,29)(30,33);(8,16)(12,14)(17,18)(19,20)

Polycyclic presentation:

2^11|a^2=f,b^2=fgjk,c^2=fj,d^2=fik,f^2=i,g^2=j,j^2=k,b^a=bg,c^a=cg,c^b=cgk,d^a=dh,d^b=dj,d^c=dijk,e^d=ei,f^b=fk,f^c=fk,g^a=gj,g^b=gk,g^c=gj,g^d=gk,j^a=jk,j^c=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319695
maximal normal subgroup of	#72319695

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