#72307222

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	#34963
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=gh,e^2=hj,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,d^a=di,d^c=dgh,e^a=ehj,e^b=ej,e^c=ei,f^a=fi,f^c=fij,f^d=fj,h^a=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307250
maximal normal subgroup of	#72307250

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