#72334838

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	#1718077
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#35390	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3010 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)(4,14)(5,17)(6,10)(8,16)(9,18)(11,22)(12,23)(13,21)(15,28)(19,30)(20,25)(24,27)(26,32)(29,31);(5,18)(7,21)(9,17)(12,19)(22,24)(31,32);(1,3)(2,7,10,21)(4,11)(5,19,18,12)(6,13)(8,24,14,22)(9,23)(15,29,20,26)(16,27)(17,30)(25,32)(28,31);(1,4)(2,8,10,14)(3,11)(5,20)(6,16)(7,22,21,24)(9,25,17,28)(12,26)(13,27)(15,18)(19,29)(23,32,30,31);(1,5)(2,9)(3,12)(4,15)(6,18)(7,23)(8,25)(10,17)(11,26)(13,19)(14,28)(16,20)(21,30)(22,31)(24,32)(27,29)

Polycyclic presentation:

2^10|c^2=f,d^2=i,b^a=bh,c^a=chj,c^b=ci,d^a=dg,d^b=dfi,d^c=df,e^a=egj,e^b=eghj,e^c=ehi,e^d=ehij,f^a=fj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334842, #72334890, #72334901, …
maximal normal subgroup of	#72334842, #72334890, #72334901, …

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