#72238249

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

82067 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=h,f^2=j,g^2=i,b^a=bhi,c^b=ch,d^a=dh,d^c=dh,f^a=fi,f^b=fhj,f^c=fh,f^d=fhj,f^e=fhij,g^b=gh,g^c=gh,g^d=gh,g^e=ghi

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340508, #72341172
maximal subgroup of	#72238707, #72315127, #72315226, …
maximal normal subgroup of	#72238707, #72315127, #72315226, …
maximal quotient of	C₂×#72238249, #72340582, #72340707, …

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