#72265945

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14885
commutator, Frattini, cosocle	C₂⁶	C₂⁴
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

4897 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:

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

Polycyclic presentation:

2^10|c^2=h,d^2=h,b^a=be,c^a=ch,c^b=cf,d^b=dg,e^c=ei,e^d=ej,f^a=fij,f^c=fj,g^a=gj,g^d=gj,h^b=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265946, #72265974
maximal normal subgroup of	#72265946, #72265974
maximal quotient of	#72266256

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