#72315221

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#532	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|c^2=f,e^2=g,g^2=i,i^2=j,b^a=bh,d^a=dj,d^b=dhj,e^c=eg,f^e=fj,g^c=gi,i^c=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315233, #72315346, #72315522
maximal normal subgroup of	#72315233, #72315346, #72315522

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