#72239150

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

41507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=j,f^2=i,g^2=j,b^a=bij,c^a=ch,c^b=chj,d^a=di,d^b=dj,e^a=ej,e^b=ehij,f^b=fi,g^a=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72239174
maximal subgroup of	#72239153, #72239174
maximal normal subgroup of	#72239153, #72239174

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