#72234339

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#33733	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=hj,b^2=eg,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=be,c^a=chj,c^b=chj,d^a=di,d^b=di,e^a=eg,e^d=ej,f^a=fj,f^b=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234355, #72260494
maximal normal subgroup of	#72234355, #72260494

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