#72265204

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#498
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	C₂×#2058
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=ej,e^2=i,f^2=j,i^2=j,b^a=bgh,c^a=cf,c^b=ce,d^c=dhj,e^a=ej,e^b=ei,g^c=gj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265233, #72307077, #72307110, …
maximal normal subgroup of	#72265233, #72307077, #72307110, …

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