#72322346

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	#7624079	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2215 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,f^2=h,g^2=i,i^2=j,b^a=bgi,c^b=cj,d^a=dhj,d^c=dhj,e^a=egj,e^d=eh,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322376, #72322909, #72343989
maximal normal subgroup of	#72322376, #72322909, #72343989
maximal quotient of	#72344008

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, 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);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
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);