#72260898

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

At least 8276 conjugacy classes of subgroups; only the 417509 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=j,b^a=bj,d^c=di,e^a=ej,e^b=eij,e^c=eij,f^d=fij,f^e=fj,g^b=gj,g^d=gj,h^a=hj,h^d=hi,h^e=hj

Conjugacy classes & character table

274 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260919
maximal normal subgroup of	#72260919

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