#72280002

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	C₂×#2804
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4931 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,d^2=hi,e^2=hj,g^2=j,b^a=bh,c^b=cfi,d^a=dg,d^b=dghij,e^a=eh,e^b=eghj,e^d=ej,f^a=fj,g^a=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280026
maximal normal subgroup of	#72280026

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