#72265486

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8074
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₄×#97	C₂³
socle	C₂	#96076
prime radical	C₂⁴	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

614 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)(3,7,14,20,13,22,16,26)(4,9,12,25)(5,8)(6,11)(10,17)(15,21,30,32,27,24,29,31)(18,23,28,19);(1,3,12,27,17,16,18,29,6,13,4,15,5,14,28,30)(2,7,19,31,8,20,9,24,11,22,23,32,10,26,25,21);(3,13)(14,16)(15,27)(29,30)

Polycyclic presentation:

2^10|a^2=e,b^2=defh,d^2=gij,e^2=hj,f^2=ij,g^2=j,i^2=j,b^a=bd,c^a=chj,c^b=cghij,d^a=dg,d^b=dgh,e^b=eij,f^a=fghij,f^b=fghi,f^d=fj,f^e=fj,g^a=gj,g^b=gj,h^b=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265508
maximal normal subgroup of	#72265508

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