#72304602

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

82907 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304735
maximal normal subgroup of	#72304735

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