#72263505

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6978 conjugacy classes of subgroups; only the 3453 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=ij,d^2=i,b^a=bh,c^a=chi,c^b=cg,d^a=dij,d^c=di,e^c=ehi,e^d=eij,f^c=fhj,f^d=fj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263549, #72328461
maximal normal subgroup of	#72263549, #72328461
maximal quotient of	#72331076, #72331185

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