#72340340

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233541	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,e^2=fgi,b^a=bf,c^a=cf,c^b=cfg,d^a=di,d^b=di,e^a=ehij,e^b=ehi,e^c=eghij,e^d=ei,f^c=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gj,g^e=gj,h^b=hj,h^c=hij,h^e=hi

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340342, #72340359, #72340380, …
maximal normal subgroup of	#72340342, #72340359, #72340380, …

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