#72265601

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15277
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,d^2=ij,e^2=j,b^a=be,c^a=cg,c^b=cgh,d^b=df,d^c=degi,e^a=ej,e^b=ej,e^d=ej,f^a=fj,g^d=gj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265627, #72313973, #72314226
maximal normal subgroup of	#72265627, #72313973, #72314226

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