#72285446

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

5709 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fhi,d^2=ij,b^a=bg,c^a=ci,c^b=cgh,d^a=dhi,d^b=dgi,d^c=dfg,e^a=ei,e^b=efhj,e^c=efhij,f^a=fj,f^c=fj,g^b=gj,g^c=gj,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

76 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.6);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);