#72284531

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#454367	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5865 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=fgij,d^2=ij,e^2=j,b^a=bi,c^a=cf,c^b=cfg,d^a=di,d^b=dh,d^c=dgij,e^b=egij,e^d=efgi,f^b=fj,f^d=fj,g^c=gj,g^d=gj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284622
maximal normal subgroup of	#72284622

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