#72312698

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14716
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6347494	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

3931 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)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,42)(29,38)(31,46)(32,40)(34,41)(35,39)(37,44)(43,47)(45,48);(1,2,4,8,5,9,14,7)(3,10,11,6,13,15,12,16)(17,19,22,30)(18,23,26,27)(20,28,32,44)(21,34)(24,37,40,42)(25,41)(29,36)(31,43)(33,38)(35,45)(39,47)(46,48);(1,3,5,13)(2,6,9,16)(4,12,14,11)(7,15,8,10)(17,20,22,32)(18,24,26,40)(19,28,30,44)(21,35)(23,37,27,42)(25,39)(29,45)(31,36)(33,46)(34,43)(38,47)(41,48);(2,7)(3,11)(4,14)(8,9)(10,15)(12,13)(17,21,22,36)(18,25,26,33)(19,29,30,34)(20,31,32,35)(23,38,27,41)(24,39,40,46)(28,43,44,45)(37,47,42,48)

Polycyclic presentation:

2^10|b^2=f,c^2=hij,d^2=i,e^2=j,f^2=j,b^a=bi,c^a=ch,c^b=cfij,d^a=di,d^b=de,d^c=deg,e^b=ei,e^c=ei,e^d=ej,f^c=fj,f^d=fij,g^a=gi,g^b=gij,g^c=gj,h^d=hi

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312723, #72312812, #72312882
maximal normal subgroup of	#72312723, #72312812, #72312882

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