#72318194

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#5228713
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,d^2=i,c^a=cg,c^b=cf,d^a=dfgh,d^b=dfi,d^c=di,e^a=eghi,e^b=egi,e^c=ehi,f^e=fj,g^d=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318234, #72319027, #72319379, …
maximal normal subgroup of	#72318234, #72319027, #72319379, …

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