#72307337

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	C₂³×C₄	C₂⁵
socle	C₂²	C₂×#2598
prime radical	#60444	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3811 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)(3,5)(4,6)(7,8)(9,10,13,20,15,19,14,17)(11,21,24,30,23,16,25,32)(12,26,28,35,27,33,29,18)(22,36,38,31,37,40,39,34);(1,3,4,7)(2,5,6,8)(9,11)(10,16,19,21)(12,22)(13,24)(14,25)(15,23)(17,30,20,32)(18,34,35,31)(26,36,33,40)(27,37)(28,38)(29,39);(10,17,19,20)(12,27)(16,30,21,32)(18,26,35,33)(22,37)(28,29)(31,40,34,36)(38,39);(3,7)(5,8)(9,12,13,28,15,27,14,29)(10,18,17,33,19,35,20,26)(11,22,24,38,23,37,25,39)(16,31,30,36,21,34,32,40);(2,6)(5,8)

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=hi,d^2=ij,h^2=j,i^2=j,b^a=bf,c^a=ci,d^a=di,d^b=dg,d^c=dj,e^a=eghi,f^b=fj,f^d=fj,g^a=gj,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307402
maximal normal subgroup of	#72307402

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