#72317212

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁴×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4381 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=eg,d^2=h,f^2=i,h^2=i,i^2=j,b^a=bg,d^a=df,d^b=dgi,d^c=dg,e^d=ej,f^a=fi,f^b=fj,h^a=hi,h^b=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317214, #72317221, #72317372
maximal normal subgroup of	#72317214, #72317221, #72317372

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