#72257693

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14715
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5523 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=g,d^2=f,e^2=ij,f^2=j,c^a=cg,c^b=cg,d^a=dh,d^b=df,d^c=de,e^a=ei,f^b=fj,f^c=fij,g^d=gij,h^c=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257748
maximal normal subgroup of	#72257748
maximal quotient of	#72257808, #72257817

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