#72326342

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 10957 conjugacy classes of subgroups; only the 3745 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,b^a=bg,c^a=cj,c^b=cgh,d^a=di,d^b=di,d^c=di,e^a=eghij,e^b=eghij,e^c=eghij,f^a=fj,f^b=fhi,f^c=fj,f^d=fhi

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326378, #72331723
maximal normal subgroup of	#72326378, #72331723
maximal quotient of	#72333326

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