#72261051

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 7748 conjugacy classes of subgroups; only the 31825 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=h,e^2=j,b^a=bi,c^b=ch,d^a=di,d^c=dhij,e^a=ej,e^d=ej,f^a=fi,f^c=fj,g^b=gi,g^c=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261078, #72261507
maximal normal subgroup of	#72261078, #72261507

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