#72319338

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	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623097	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2509 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=fh,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=cghi,d^b=dij,e^b=ej,f^b=fj,g^c=gi,g^d=gj,i^c=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319381, #72319510, #72319631, …
maximal normal subgroup of	#72319381, #72319510, #72319631, …
maximal quotient of	#72343903

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