#72285204

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
prime radical	#394061	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5535 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:

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

Polycyclic presentation:

2^10|b^2=fg,c^2=j,d^2=hi,e^2=hi,b^a=bf,c^a=cj,c^b=cg,d^a=dh,d^b=dj,d^c=dgh,e^a=ehj,e^b=ehij,e^c=eg,e^d=ej,f^d=fj,f^e=fj,g^d=gj,g^e=gj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285318
maximal normal subgroup of	#72285318

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