#72267321

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄³

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

166448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^11|e^2=k,f^2=k,c^a=cg,c^b=ch,d^a=dgi,d^b=dik,e^c=ej,e^d=ek,f^c=fk,f^d=fjk,f^e=fk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267330, #72267346
Fitting subgroup of	#72267330
maximal subgroup of	#72267324, #72267330
maximal normal subgroup of	#72267324, #72267330

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