#72267584

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

92889 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:

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

Polycyclic presentation:

2^10|a^2=i,f^2=h,b^a=bh,c^a=chij,d^a=dhij,e^a=eh,f^a=fhi,f^b=fij,f^c=fhij,f^d=fhij,f^e=fh,g^a=gj,g^b=ghij,g^c=ghij,g^e=gh

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267594
maximal normal subgroup of	#72267594
maximal quotient of	#72330093, #72330120, #72330243, …

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