#72343005

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	#7623513	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2399 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:

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

Polycyclic presentation:

2^10|a^2=f,f^2=h,g^2=i,i^2=j,b^a=bi,c^b=ch,e^a=eg,e^b=eg,e^c=eh,e^d=eh,f^e=fj,g^a=gi,g^b=gij,g^e=gij,i^a=ij,i^b=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343057
maximal normal subgroup of	#72343057
maximal quotient of	#72343341, #72343582, #72343805, …

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