#72265607

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	C₂×C₄	#2557
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	#15270
prime radical	C₂²×#575	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5767 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=i,d^2=h,e^2=j,b^a=be,c^b=cegi,d^a=dg,d^b=dh,d^c=df,e^a=ej,e^b=ej,e^c=ej,f^a=fj,g^c=gj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265627, #72314015, #72314313
maximal normal subgroup of	#72265627, #72314015, #72314313

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