#72255772

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#119484
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,g^2=j,b^a=bg,c^a=ce,c^b=cdj,d^b=di,e^b=ej,f^a=fh,f^c=fi,g^a=gij,g^b=ghj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255779, #72255834, #72255880, …
maximal normal subgroup of	#72255779, #72255834, #72255880, …

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