#72319311

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	#7623099	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2400 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=f,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=cj,c^b=cghj,d^b=di,d^c=dhj,e^b=ehj,e^d=eh,f^b=fj,g^c=gi,g^d=gj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319381, #72319390, #72319621
maximal normal subgroup of	#72319381, #72319390, #72319621

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