#72263827

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=hij,d^2=fgj,b^a=bj,c^a=cehi,c^b=chi,d^a=df,d^b=dj,d^c=dg,e^d=ei,f^a=fi,f^c=fh,f^d=fh,f^e=fj,g^a=ghi,g^d=gh,g^f=gj,h^a=hj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263829, #72263836
maximal normal subgroup of	#72263829, #72263836

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