#72334804

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	#2827566
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#34980	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3644 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)(4,8,15,23)(5,9)(6,10)(11,18,28,21)(12,19)(13,20)(14,22,16,25)(17,26)(24,30)(27,31,29,32);(1,3)(2,7)(4,11)(5,12,17,30)(6,13)(8,21)(9,24,26,19)(10,20)(14,29,16,27)(15,28)(18,23)(22,32,25,31);(1,4)(2,8,10,23)(3,11)(5,16,17,14)(6,15)(7,18,20,21)(9,25)(12,29,30,27)(13,28)(19,32)(22,26)(24,31);(1,5)(2,9)(3,12)(4,14)(6,17)(7,19)(8,22)(10,26)(11,27)(13,30)(15,16)(18,31)(20,24)(21,32)(23,25)(28,29);(7,20)(18,21)(19,24)(31,32)

Polycyclic presentation:

2^10|a^2=h,b^2=fh,c^2=fhi,b^a=bf,c^a=chi,c^b=cfhi,d^b=dg,d^c=dh,e^a=eghij,e^b=ei,f^c=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334806, #72334823, #72334833, …
maximal normal subgroup of	#72334806, #72334823, #72334833, …

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