#72319196

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#4893962
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3123 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)(4,8,15,22)(5,16,17,9)(6,10)(11,18,25,31)(12,19,27,29)(13,20)(14,21)(23,28)(24,32)(26,30);(1,3)(2,7)(4,11)(5,12,17,27)(6,13)(8,18)(9,19,16,29)(10,20)(14,26,28,24)(15,25)(21,32,23,30)(22,31);(1,4)(2,8,10,22)(3,11)(5,14)(6,15)(7,18,20,31)(9,23,16,21)(12,26)(13,25)(17,28)(19,30,29,32)(24,27);(1,5)(2,9)(3,12)(4,14)(6,17)(7,19)(8,21)(10,16)(11,24)(13,27)(15,28)(18,30)(20,29)(22,23)(25,26)(31,32);(7,20)(18,31)(19,29)(30,32)

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=i,b^a=bf,c^a=cfgi,d^a=dfgh,d^b=dfghij,d^c=dh,e^a=ehi,e^b=ei,f^d=fj,g^c=gj,g^d=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319229, #72319408, #72321499, …
maximal normal subgroup of	#72319229, #72319408, #72321499, …

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