#72284382

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁵
prime radical	#35272	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,c^2=h,d^2=j,f^2=h,b^a=bf,c^a=ch,d^a=dh,d^b=dhj,e^b=eg,e^c=eh,e^d=ej,f^a=fh,f^e=fi,g^a=gij,g^b=gi,g^f=gj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284567, #72285086, #72285162
maximal normal subgroup of	#72284567, #72285086, #72285162

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