#72285685

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1807
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#41	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=g,e^2=h,f^2=hij,g^2=i,i^2=j,b^a=bgij,c^b=cf,d^b=dh,e^a=ej,e^c=ej,f^b=fhij,f^c=fhj,g^b=gi,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286063
maximal normal subgroup of	#72286063

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