#72307022

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=ghj,c^2=fhj,e^2=i,f^2=ij,i^2=j,b^a=bg,c^a=ce,c^b=cefi,d^c=dgh,e^a=ei,e^c=ej,f^a=fij,f^b=fj,f^c=fj,g^c=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307029, #72307085, #72307248
maximal normal subgroup of	#72307029, #72307085, #72307248

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