#72287162

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	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2330 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=fhij,f^2=h,g^2=i,i^2=j,c^a=chj,d^a=dhj,d^b=dhj,d^c=dg,e^d=eghj,f^d=fj,g^c=gi,g^d=gi,g^e=gij,i^c=ij,i^d=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287751, #72318672
maximal normal subgroup of	#72287278, #72287751, #72318672

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