#72318318

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	#2539743
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#395133	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3307 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)(3,11)(4,15)(5,9)(6,10)(7,14)(8,18)(12,25)(13,26)(16,21)(17,23)(19,30)(20,24)(22,29)(27,31)(28,32);(1,3,6,14)(2,7)(4,16,18,12)(5,13,20,29)(8,21)(9,22)(10,11)(15,25)(17,27,19,28)(23,32)(24,26)(30,31);(1,4,6,18)(2,8,10,15)(3,12)(5,19,20,17)(7,21)(9,23,24,30)(11,25)(13,28)(14,16)(22,31)(26,32)(27,29);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,23)(10,24)(11,26)(12,27)(14,29)(15,30)(16,28)(18,19)(21,31)(25,32);(5,20)(13,29)(17,19)(27,28)

Polycyclic presentation:

2^10|b^2=i,c^2=hj,b^a=bh,c^a=chj,c^b=cf,d^a=dfg,d^b=dfg,d^c=di,e^a=egij,e^d=ei,f^a=fj,f^c=fj,f^d=fj,g^a=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318468, #72319065, #72319168, …
maximal normal subgroup of	#72318468, #72319065, #72319168, …

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