#72319652

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1926
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#60673	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20473 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=ei,c^2=efhj,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=cf,d^c=dhj,e^b=ej,f^a=fh,f^b=fj,g^a=gi,g^b=gi,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319682, #72321935
maximal normal subgroup of	#72319682, #72321935

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