#72307046

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=h,f^2=i,i^2=j,b^a=bf,c^a=cg,c^b=cgi,d^b=dgh,e^c=ej,f^a=fi,g^b=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307057, #72307077, #72307248
maximal normal subgroup of	#72307057, #72307077, #72307248

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