#72306602

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6165 conjugacy classes of subgroups; only the 29760 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^11|e^2=k,b^a=bj,c^b=ch,d^a=dh,d^b=dj,e^a=ek,e^b=ej,e^c=ei,e^d=ek,f^a=fi,g^d=gj,g^f=gk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306611, #72324510, #72324639
maximal normal subgroup of	#72306611, #72324510, #72324639

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