#72335975

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

40691 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|a^2=i,b^2=ij,d^2=ghi,e^2=ghj,f^2=g,b^a=bg,c^a=cij,c^b=cij,d^a=dgh,d^b=dghj,d^c=dij,e^a=eghij,e^b=eghi,e^c=eij,e^d=eij,f^a=fgj,f^d=fgi,f^e=fi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336001, #72336147
maximal normal subgroup of	#72336001, #72336147
maximal quotient of	#72336474

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