#72239741

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#379
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#7712063	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

4918 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)(25,26,29,32)(27,35,38,31)(28,33,30,34)(36,40,37,39);(1,3)(2,4,7,8)(5,6)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22)(25,27,28,37,29,38,30,36)(26,31,33,40,32,35,34,39);(1,4)(2,6,7,3)(5,8)(10,15,16,20)(14,17,23,24)(27,36,38,37)(31,39,35,40);(10,16)(11,18)(15,20)(19,21)(26,32)(31,35)(33,34)(39,40)

Polycyclic presentation:

2^10|a^2=hj,b^2=e,c^2=egi,e^2=j,f^2=h,g^2=j,b^a=bf,c^a=cfhij,c^b=cg,d^a=dhj,d^b=dh,e^a=ei,e^c=ei,f^a=fh,f^b=fhi,f^c=fi,g^a=gh,g^b=gij,g^c=gi

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239752
maximal normal subgroup of	#72239752

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