#72315027

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

69643 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)(8,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,27)(24,26)(25,28);(1,2)(3,4)(5,6,9,10)(7,8,11,12)(13,14,18,21)(15,22,17,19)(16,24,26,20)(23,27,25,28);(7,11)(8,12)(13,15,18,17)(14,19,21,22)(16,23,26,25)(20,27,24,28);(1,2,3,4);(1,2)(3,4)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,24)(22,28);(5,7)(6,8)(9,11)(10,12)(13,17,18,15)(14,19,21,22)(16,25,26,23)(20,27,24,28);(5,8)(6,7)(9,12)(10,11)(13,14)(15,19)(16,20)(17,22)(18,21)(23,27)(24,26)(25,28)

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=j,d^2=h,f^2=j,b^a=bhi,c^b=cj,d^b=dh,e^a=eh,e^b=ej,e^d=eh,f^a=fj,f^c=fij,g^a=gij,g^b=gi,g^c=gij,g^f=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315043, #72315226
maximal normal subgroup of	#72315043, #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);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.8);;
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.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, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);