#72249482

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 12502 conjugacy classes of subgroups; only the 31809 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)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(2,4)(7,11)(10,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(16,24)(19,26)(22,28)(25,27);(2,4)(6,9)(10,12)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28)

Polycyclic presentation:

2^10|b^2=i,f^2=j,b^a=bi,c^a=ci,e^c=ej,e^d=ei,f^a=fij,f^b=fh,f^e=fhj,g^b=gi,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567, #72256240
maximal normal subgroup of	#72249567, #72256240

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