#72321172

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#3444	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2212 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=hj,f^2=h,g^2=i,i^2=j,b^a=bgij,d^a=dj,e^a=ej,e^c=ej,e^d=eh,f^a=fj,g^b=gi,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321406, #72322136
maximal normal subgroup of	#72321406, #72322136
maximal quotient of	#72343794

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