#72285228

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

5725 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)(9,21)(11,24)(12,27)(16,18)(19,23)(26,34)(28,29);(1,2,3,4)(5,6)(7,11)(8,12,20,27)(9,22,21,13)(10,14)(15,24)(16,25,18,33)(17,26,30,34)(19,28)(23,29)(31,36)(32,35);(5,7,10,15)(6,11,14,24)(8,18)(9,17)(12,25)(13,26)(16,20)(19,32,23,31)(21,30)(22,34)(27,33)(28,35,29,36);(5,8,10,20)(6,12)(7,16,15,18)(9,23,21,19)(11,25)(13,29)(14,27)(17,32,30,31)(22,28)(24,33)(26,36)(34,35);(5,9,10,21)(6,13)(7,17,15,30)(8,19,20,23)(11,26)(12,28)(14,22)(16,31,18,32)(24,34)(25,35)(27,29)(33,36)

Polycyclic presentation:

2^10|b^2=f,c^2=gj,d^2=hj,e^2=hj,b^a=bf,c^a=cj,c^b=cg,d^a=dh,d^b=dgh,d^c=di,e^a=ehj,e^b=egij,e^c=ehij,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

82 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);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
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.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
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);