#72309682

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1879
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#7623358	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efj,c^2=efgh,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bf,c^a=cfgij,c^b=cfi,d^a=dhj,d^c=dh,e^b=eh,e^c=ehj,f^b=fh,f^c=fh,g^a=gi,g^b=gj,g^c=gj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309687
maximal normal subgroup of	#72309687

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