#72318815

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	#3444	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2481 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bj,c^a=chij,c^b=ci,d^a=dj,d^c=dj,e^b=ej,e^d=ehi,f^c=fj,g^b=gj,g^c=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318841, #72318898, #72319058, …
maximal normal subgroup of	#72318841, #72318898, #72319058, …

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