#72285240

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=i,d^2=ij,c^a=cg,c^b=cf,d^a=di,d^b=dh,d^c=dfghij,e^a=ej,e^b=egh,e^c=efhij,e^d=efhij,f^b=fj,g^d=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285704
maximal normal subgroup of	#72285704

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