#72268633

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂³	#2804
prime radical	C₂⁵	H(5,𝔽₂)
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=egh,c^2=hi,d^2=ghi,f^2=j,b^a=beg,c^a=chi,c^b=cf,d^a=dfhj,d^b=dfgi,d^c=di,e^b=ej,f^a=fj,f^b=fj,g^a=gj,h^b=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268636
maximal normal subgroup of	#72268636

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