#72300569

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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 7731 conjugacy classes of subgroups; only the 3829 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=gh,b^a=bi,c^a=cg,d^a=dghj,d^c=dghj,e^a=eh,e^b=eghj,e^c=eij,e^d=eghj,f^a=fh,f^b=fghj,f^c=fgj,f^d=fghj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300575, #72318003, #72323275, …
maximal normal subgroup of	#72300575, #72318003, #72323275, …
maximal quotient of	#72323724, #72323727, #72323730, …

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