#72319225

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	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623097	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2306 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=fi,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=cg,d^a=dh,d^b=dhij,d^c=dh,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319265, #72319631, #72319699
maximal normal subgroup of	#72319265, #72319631, #72319699

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