#72233765

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

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

At least 7285 conjugacy classes of subgroups; only the 31937 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 20 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,19)(18,20);(5,7,10,12)(6,11,8,9);(2,4)(5,6)(7,9)(8,10)(11,12)(15,20)(17,18);(5,7,10,12)(6,11,8,9)(13,15)(14,17)(16,20)(18,19);(6,8)(9,11)(13,15)(14,18)(16,20)(17,19);(1,2,3,4)(5,8,10,6)(7,9,12,11)(13,14)(15,17)(16,19)(18,20);(5,9)(6,7)(8,12)(10,11)

Polycyclic presentation:

2^10|b^2=i,d^2=i,f^2=h,b^a=bi,c^b=ci,d^a=di,d^c=dij,e^a=eij,e^c=eij,f^a=fh,f^c=fh,f^e=fij,g^e=gi

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234468, #72238707
maximal normal subgroup of	#72234468, #72238707

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