#72233892

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13967
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂⁶	C₂⁴
prime radical	C₂×#23900	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

10332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=i,d^2=j,b^a=bef,c^a=cg,c^b=cefgij,d^a=dh,d^b=dij,d^c=dj,e^c=ei,e^d=ej,f^c=fi,g^a=gi,g^b=gi,g^d=gj,h^a=hj,h^b=hj,h^c=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233899, #72234565, #72234571
maximal normal subgroup of	#72233899, #72234565, #72234571

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