#72238786

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3483 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=fgij,d^2=h,e^2=j,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bh,c^a=cf,c^b=cgj,d^a=dij,d^b=dj,e^a=ehij,e^c=ehi,f^a=fi,f^c=fj,g^c=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238791, #72239846, #72239912
maximal normal subgroup of	#72238791, #72239846, #72239912

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