#72303306

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
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

51505 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)(9,10)(11,13)(12,16)(14,15)(17,18,22,20)(19,24,21,23);(1,2)(3,4)(5,6)(7,8)(9,11,14,16)(10,12,15,13)(17,19,22,21)(18,23,20,24);(2,4)(5,6,7,8)(10,15)(12,13)(18,20)(19,21);(1,2)(3,4)(5,6)(7,8)(9,12,14,13)(10,11,15,16)(17,20,22,18)(19,23,21,24);(9,11)(10,12)(13,15)(14,16);(1,2)(3,4)(5,6)(7,8)(9,13,14,12)(10,16,15,11)(17,21,22,19)(18,24,20,23)

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=j,d^2=gh,f^2=gh,b^a=bgh,c^a=ci,c^b=chij,d^a=dh,d^b=dg,d^c=dij,e^a=eh,e^b=eh,e^d=eh,f^a=fgh,f^c=fij,f^d=fg,f^e=fh

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303323
maximal normal subgroup of	#72303323

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