#72323019

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	#7623114	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17316 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2416 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=h,f^2=h,g^2=i,i^2=j,b^a=bghi,c^a=ch,c^b=ch,d^a=dgh,d^c=dh,e^c=ej,f^a=fj,g^b=gi,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343989
maximal normal subgroup of	#72343989

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