#72317871

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	#2618285
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#34980	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3594 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)(5,9)(6,10)(11,22,28,19)(12,20,16,25)(14,23)(15,24)(17,30)(18,26)(27,32,29,31);(1,3)(2,7,10,21)(4,11)(5,16,18,12)(6,13)(8,22,24,19)(9,25)(14,27,17,29)(15,28)(20,26)(23,32)(30,31);(1,4)(2,8)(3,11)(5,17)(6,15)(7,19)(9,23)(10,24)(12,27)(13,28)(14,18)(16,29)(20,32)(21,22)(25,31)(26,30);(1,5)(2,9)(3,12)(4,14)(6,18)(7,20)(8,23)(10,26)(11,27)(13,16)(15,17)(19,31)(21,25)(22,32)(24,30)(28,29);(8,24)(19,22)(23,30)(31,32)

Polycyclic presentation:

2^10|a^2=h,b^2=f,b^a=bhi,c^a=cfhi,c^b=ci,d^b=dg,d^c=dhij,e^a=egh,e^c=ei,f^a=fj,f^d=fj,g^b=gj,g^c=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317935, #72319027, #72319065, …
maximal normal subgroup of	#72317935, #72319027, #72319065, …

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