#72308396

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#394
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₄×#394
prime radical	#60900	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,12,15)(3,9,11,13,10,14,16,6)(17,18,21,27,23,28,22,24)(19,29,31,25,33,39,32,41)(20,34,37,44,35,42,38,26)(30,43,46,48,36,40,47,45);(4,12)(6,13)(7,15)(11,16)(18,24,28,27)(20,35)(25,39,41,29)(26,42,44,34)(30,36)(37,38)(40,45,43,48)(46,47);(1,3)(2,6,8,13)(4,11)(5,10)(7,9,15,14)(12,16)(17,19)(18,25)(20,36)(21,32)(22,31)(23,33)(24,39)(26,43)(27,29)(28,41)(30,35)(34,45)(37,46)(38,47)(40,44)(42,48);(2,7,8,15)(3,10)(6,14,13,9)(11,16)(17,20,22,37,23,35,21,38)(18,26,24,34,28,44,27,42)(19,30,31,47,33,36,32,46)(25,40,39,48,41,43,29,45)

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=ghi,d^2=fgi,e^2=ij,g^2=j,h^2=j,b^a=bg,c^a=ceghj,c^b=cghi,d^a=deghi,d^b=dghij,d^c=dij,e^b=ei,e^c=ej,e^d=ej,f^a=fj,g^a=gij,h^a=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308460
maximal normal subgroup of	#72308460

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