#72285133

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

5438 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)(11,22)(13,28)(17,31)(19,34)(24,29)(26,30);(1,2,3,4)(5,6)(7,11)(8,12,20,27)(9,13,21,28)(10,14)(15,22)(16,25,18,23)(17,24,31,29)(19,26)(30,34)(32,36)(33,35);(5,7,10,15)(6,11,14,22)(8,18)(9,17)(12,25)(13,29)(16,20)(19,32,34,33)(21,31)(23,27)(24,28)(26,36,30,35);(5,8,10,20)(6,12)(7,16,15,18)(9,19,21,34)(11,23)(13,30)(14,27)(17,33,31,32)(22,25)(24,35)(26,28)(29,36);(5,9)(6,13)(7,17,15,31)(8,19)(10,21)(11,24,22,29)(12,26)(14,28)(16,32,18,33)(20,34)(23,35,25,36)(27,30)

Polycyclic presentation:

2^10|b^2=fg,c^2=gj,d^2=hij,e^2=h,b^a=bf,c^a=cgj,c^b=cg,d^a=dh,d^b=dgi,d^c=dh,e^a=ehj,e^b=ehij,e^c=egi,e^d=ei,g^d=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

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