#72334873

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	#5782689
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#2021	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2941 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)(3,7,13,21)(4,8)(5,16,18,9)(6,10)(11,22,28,19)(12,25)(14,26,29,23)(15,24)(17,20)(27,31)(30,32);(1,3)(2,7,10,21)(4,11)(5,17,18,12)(6,13)(8,22,24,19)(9,25)(14,27,29,30)(15,28)(16,20)(23,32)(26,31);(1,4)(2,8)(3,11)(5,14,18,29)(6,15)(7,19)(9,26,16,23)(10,24)(12,27,17,30)(13,28)(20,32,25,31)(21,22);(1,5)(2,9,10,16)(3,12)(4,14,15,29)(6,18)(7,20,21,25)(8,23)(11,27,28,30)(13,17)(19,31)(22,32)(24,26);(8,24)(19,22)(23,26)(31,32)

Polycyclic presentation:

2^10|a^2=g,b^2=fg,c^2=fgi,d^2=h,b^a=bf,c^a=cg,c^b=cfg,d^a=dfgj,d^b=dfh,d^c=dfghi,e^a=ehi,e^c=ei,f^a=fj,f^b=fj,g^b=gj,g^d=gj,h^a=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334879, #72334880, #72334895, …
maximal normal subgroup of	#72334879, #72334880, #72334895, …

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