#72339684

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	#1958285
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#393979	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4123 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=ghj,g^2=j,h^2=j,b^a=bg,c^a=cfhj,c^b=cfgij,d^b=df,d^c=dij,e^a=eij,e^b=ehj,e^c=ehi,e^d=ej,f^c=fj,g^a=gj,g^b=gj,h^a=hj,h^b=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339689
maximal normal subgroup of	#72339689

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