#72343030

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	C₂²×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2351 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,f^2=h,g^2=i,i^2=j,b^a=bg,d^b=dh,e^c=eh,f^b=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343057
maximal normal subgroup of	#72343057
maximal quotient of	#72343794

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