#72321844

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ C₂×#2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14)(15,21)(16,22)(17,30)(18,24)(19,34)(20,27)(23,33)(25,38)(26,39)(28,40)(29,41)(31,36)(32,44)(35,42)(37,43);(6,9)(7,11)(13,15)(14,20)(16,26)(17,31)(18,25)(19,28)(21,30)(22,36)(23,40)(24,35)(27,33)(29,42)(32,43)(34,37)(38,44)(39,41);(13,15)(14,21)(16,28)(17,27)(18,25)(19,26)(20,30)(22,40)(23,36)(24,38)(29,43)(31,33)(32,42)(34,39)(35,44)(37,41);(2,4)(5,6,8,9)(7,12,11,10)(15,25)(20,35)(21,38)(26,28)(27,42)(31,43)(36,37)(39,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,22,24,34)(15,26,25,28)(17,29,32,33)(20,36,35,37)(21,39,38,40)(23,30,41,44)(27,31,42,43)

Polycyclic presentation:

2^10|d^2=hj,e^2=h,g^2=ij,i^2=j,b^a=bg,d^a=df,d^b=dh,d^c=dj,e^a=ef,e^b=ehj,e^c=ej,e^d=ehj,g^a=gi,g^b=gi,i^a=ij,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321911
maximal normal subgroup of	#72321911

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