#72304639

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

81866 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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=i,f^2=j,g^2=ij,c^b=cij,f^b=fj,f^c=fj,f^d=fj,f^e=fh,g^a=gi,g^b=gj,g^c=gi,g^d=gij,g^e=ghi

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