#72342338

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

88693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 28 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,21)(15,20,24,28)(16,22,26,25)(17,23,27,18);(2,3)(5,6)(7,9)(8,11)(10,12)(13,15)(14,20)(16,17)(18,25)(19,24)(21,28)(22,23)(26,27);(1,2,4,3)(6,7)(10,11)(13,14)(15,20)(16,25)(17,18)(19,21)(22,26)(23,27)(24,28);(1,3,4,2)(5,7,9,6)(8,11,12,10)(14,21)(16,26)(17,27)(20,28);(13,16)(14,22)(15,17)(18,28)(19,26)(20,23)(21,25)(24,27);(13,17)(14,23)(15,16)(18,21)(19,27)(20,22)(24,26)(25,28);(5,8,9,12)(6,10,7,11)(13,18,19,23)(14,17,21,27)(15,22,24,25)(16,20,26,28)

Polycyclic presentation:

2^10|a^2=h,c^2=hi,d^2=j,g^2=ij,c^a=chj,c^b=cj,d^a=dhj,d^b=dj,d^c=dij,e^c=eh,e^d=eh,f^c=fh,f^d=fh,g^b=gij,g^e=gh

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

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