#72251157

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#4387585
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4813 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)(4,14,17,8)(5,9)(6,10)(11,21,15,30)(12,28)(13,23)(16,25,19,24)(18,22)(20,26)(27,32,29,31);(1,3)(2,7)(4,15,17,11)(5,18)(6,13)(8,21,14,30)(9,22)(10,23)(12,20)(16,29,19,27)(24,32,25,31)(26,28);(1,4)(2,8)(3,11,13,15)(5,19)(6,17)(7,21,23,30)(9,25)(10,14)(12,29,18,27)(16,20)(22,32,28,31)(24,26);(1,5)(2,9)(3,12)(4,16)(6,20)(7,22)(8,24)(10,26)(11,27)(13,18)(14,25)(15,29)(17,19)(21,31)(23,28)(30,32);(5,20)(8,14)(11,15)(22,28)

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=g,c^a=cfi,c^b=cfh,d^a=dg,d^b=dgij,d^c=dj,e^a=ehij,e^b=ehij,e^c=ehj,e^d=eij,f^c=fj,g^b=gj,h^a=hj,h^b=hj,h^d=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251165, #72251291, #72251313, …
maximal normal subgroup of	#72251165, #72251291, #72251313, …

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