#72315015

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

67617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=ij,c^2=h,d^2=hj,f^2=j,b^a=bhi,c^b=chj,d^b=dh,e^a=eh,e^b=ej,e^c=eh,e^d=eh,f^a=fij,f^c=fi,g^b=gij,g^c=gi,g^f=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315023, #72315226
maximal normal subgroup of	#72315023, #72315226

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