#72342348

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 7044 conjugacy classes of subgroups; only the 31925 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=h,d^2=hij,e^2=hij,b^a=bij,c^a=ch,c^b=cij,d^a=dhj,d^b=dij,d^c=dj,e^a=ehij,e^b=eij,e^c=eij,e^d=ei,f^a=fhi,f^c=fhi,f^d=fhi,f^e=fhi,g^b=gj,g^c=gi,g^d=gj,g^e=ghij,g^f=gi

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