#72334855

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	#3066521
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#422795	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3434 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)(4,8,15,24)(5,9)(6,10)(11,22,28,20)(12,29)(13,19)(14,23,17,25)(16,21)(18,26)(27,31,30,32);(7,19)(14,17)(20,22)(21,29)(23,25)(27,30);(1,3)(2,7,10,19)(4,11)(5,16,18,12)(6,13)(8,22,24,20)(9,21)(14,30,17,27)(15,28)(23,32)(25,31)(26,29);(1,4)(2,8,10,24)(3,11)(5,17)(6,15)(7,20,19,22)(9,25,26,23)(12,27)(13,28)(14,18)(16,30)(21,31,29,32);(1,5)(2,9)(3,12)(4,14)(6,18)(7,21)(8,23)(10,26)(11,27)(13,16)(15,17)(19,29)(20,31)(22,32)(24,25)(28,30)

Polycyclic presentation:

2^10|a^2=g,c^2=f,d^2=i,b^a=bh,c^a=chj,c^b=ci,d^a=dghi,d^b=dfi,d^c=df,e^a=eh,e^b=eg,e^c=ehi,e^d=ehj,f^a=fj,f^e=fj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334856, #72334890, #72334892, …
maximal normal subgroup of	#72334856, #72334890, #72334892, …

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