#72318541

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3897785
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3707 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)(3,11)(4,15,17,8)(5,9)(6,10)(7,14)(12,24,27,19)(13,25)(16,23,30,21)(18,22)(20,28)(26,32,29,31);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,19,15,24)(9,20,22,25)(16,29)(17,27)(18,28)(21,32,23,31)(26,30);(5,18)(9,22)(16,30)(21,23);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,24)(9,23,22,21)(13,26)(14,27)(18,30)(20,32,25,31)(28,29);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,22)(11,25)(12,26)(14,28)(15,23)(17,30)(19,31)(24,32)(27,29)

Polycyclic presentation:

2^10|a^2=g,b^2=i,d^2=i,b^a=bgh,c^b=cf,d^a=dfghij,d^b=df,e^b=eg,e^c=ehj,e^d=ei,f^e=fj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318710, #72319229, #72334815, …
maximal normal subgroup of	#72318710, #72319229, #72334815, …

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