#72310148

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623356	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gh,c^2=fj,e^2=hi,g^2=i,i^2=j,b^a=bg,c^b=chij,d^a=dj,d^b=dh,e^a=ehi,e^c=ej,f^b=fj,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310150
maximal normal subgroup of	#72310150

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