#72322352

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#3444	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2210 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=cj,c^b=cj,d^a=dj,e^c=ej,e^d=eh,f^a=fj,g^b=gi,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322376, #72322669
maximal normal subgroup of	#72322376, #72322669

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