#72287154

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

2484 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,18,21,22,23,26,38,39)(19,27,29,41,31,30,32,24)(20,33,35,42,34,36,37,25)(28,44,46,43,45,40,48,47);(17,19,22,30,21,32,26,41,23,31,39,27,38,29,18,24)(20,28,36,47,37,46,42,44,34,45,33,43,35,48,25,40);(2,7)(6,9)(8,14)(13,15)(17,20,18,25,38,35,39,33,23,34,26,42,21,37,22,36)(19,28,24,40,29,48,27,43,31,45,41,44,32,46,30,47);(3,10)(6,9)(11,16)(13,15)(20,34)(25,42)(28,45)(33,36)(35,37)(40,44)(43,47)(46,48)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=gij,f^2=h,g^2=i,i^2=j,b^a=bhj,c^b=cij,d^a=dhj,d^b=dij,e^b=ehj,e^d=ej,f^c=fj,f^d=fj,g^b=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287214, #72287431, #72287432, …
maximal normal subgroup of	#72287214, #72287431, #72287432, …

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