#72343009

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	#7623520	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2395 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)(17,18,22,27,23,28,20,25)(19,24,29,36,31,38,32,39)(21,26,35,42,34,41,33,40)(30,37,43,46,44,47,45,48);(1,2)(3,11)(4,6)(5,13)(7,9)(8,10)(14,16)(17,19)(18,24)(20,32)(21,30)(22,29)(23,31)(25,39)(26,37)(27,36)(28,38)(33,45)(34,44)(35,43)(40,48)(41,47)(42,46);(17,20,23,22)(18,25,28,27)(19,29,31,32)(21,33,34,35)(24,36,38,39)(26,40,41,42)(30,43,44,45)(37,46,47,48);(17,21)(18,26)(19,30)(20,33)(22,35)(23,34)(24,37)(25,40)(27,42)(28,41)(29,43)(31,44)(32,45)(36,46)(38,47)(39,48);(1,3)(2,8)(4,14)(5,15)(6,11)(7,10)(9,12)(13,16)(21,34)(26,41)(30,44)(33,35)(37,47)(40,42)(43,45)(46,48)

Polycyclic presentation:

2^10|a^2=f,c^2=h,f^2=h,g^2=i,i^2=j,b^a=bi,c^b=ch,e^a=eg,e^b=eg,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	#72343379, #72343570

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, 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.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, 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);