#72308647

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂²×#459	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ #13856 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=g,d^2=gj,e^2=hi,i^2=j,c^a=cf,c^b=cf,d^a=df,d^b=dfgh,d^c=dghj,e^a=ei,e^b=eghj,e^c=egj,e^d=eg,i^a=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308733
maximal normal subgroup of	#72308733

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