#72285591

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#121
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂×C₄×#121
prime radical	#6353916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=g,d^2=f,e^2=hj,f^2=i,g^2=ij,h^2=j,b^a=bg,c^a=cij,c^b=ci,d^b=dh,e^a=eij,e^b=ehij,e^d=ej,f^b=fj,g^a=gij,h^b=hj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285790, #72286063
maximal normal subgroup of	#72285790, #72286063

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