#72237663

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×C₂²⋊C₄
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

48837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=g,d^2=f,e^2=j,b^a=bi,d^b=dh,e^c=ej,f^b=fj,g^d=gj,h^b=hj,h^d=hj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237673, #72251085, #72251214, …
maximal normal subgroup of	#72237673, #72251085, #72251214, …
maximal quotient of	#72251168, #72256571

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