#72239554

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7622935	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4233 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,d^2=ghij,e^2=hij,g^2=i,h^2=j,i^2=j,d^a=dg,d^b=dh,e^c=ej,e^d=ehij,f^d=fj,g^a=gi,g^b=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239561, #72264884, #72320956
maximal normal subgroup of	#72239561, #72264884, #72320956

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