#72333464

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8816 conjugacy classes of subgroups; only the 3273 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=ghj,e^2=h,b^a=bg,c^b=ch,d^a=dghij,d^b=dgj,d^c=dghij,e^a=ehi,e^b=egj,e^c=ehi,e^d=egj,f^b=fghj,f^c=fi,f^d=fh,f^e=fgj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333496
maximal normal subgroup of	#72333496

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