#72315107

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

83201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|d^2=i,f^2=ij,g^2=j,e^a=eij,e^b=eh,e^c=ehj,f^b=fij,f^c=fij,f^d=fi,f^e=fij,g^f=gj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315114
maximal normal subgroup of	#72315114

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