#72309716

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23188
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33717	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8282 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=e,d^2=gj,f^2=h,g^2=i,i^2=j,c^b=cfj,d^a=dg,d^b=dh,d^c=dhi,e^d=ej,f^c=fh,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309744, #72310092
maximal normal subgroup of	#72309744, #72310092

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