#72239635

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dej,c^2=eghj,d^2=i,e^2=j,f^2=i,h^2=j,b^a=bf,c^a=cfg,c^b=ch,d^b=dj,e^a=ej,e^b=ej,f^a=fij,f^b=fij,g^a=gj,g^c=gj,h^a=hij,h^b=hj,h^c=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239793, #72239917
maximal normal subgroup of	#72239793, #72239917
maximal quotient of	#72239642, #72239879

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