#72287367

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×#60695	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4970 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:

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

Polycyclic presentation:

2^10|d^2=f,e^2=fghi,f^2=h,g^2=i,i^2=j,c^a=cj,d^a=dj,d^b=dh,d^c=dhj,e^a=ej,e^b=eh,e^d=eg,f^e=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287390, #72287479, #72320918
maximal normal subgroup of	#72287390, #72287479, #72320918

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