#72257348

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊#95
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₂×C₂²⋊#95
prime radical	C₂³×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=h,e^2=i,f^2=i,h^2=j,b^a=bg,c^b=ce,d^b=di,e^a=ei,e^b=ej,f^c=fij,g^c=gi,h^b=hi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257390
maximal normal subgroup of	#72257390
maximal quotient of	#72257866

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