#72322423

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=cj,c^b=cj,d^a=dh,d^b=dj,e^a=eghj,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322467
maximal normal subgroup of	#72322467

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