#72285657

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1786
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#61431	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=e,c^2=efhij,e^2=hi,f^2=hi,g^2=i,h^2=j,b^a=bg,c^a=cg,c^b=cf,d^a=di,d^c=dij,e^c=eh,f^a=fi,f^b=fij,f^c=fh,g^a=gi,g^b=gi,h^c=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285790
maximal normal subgroup of	#72285790

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