#72263834

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6491103	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=hi,c^2=ehij,d^2=egj,b^a=bhij,c^a=chij,c^b=ce,d^a=dhi,d^b=deg,d^c=df,e^d=ei,f^b=fh,f^c=fi,f^d=fhij,f^e=fj,g^c=ghij,g^d=gi,g^f=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263836, #72316525
maximal normal subgroup of	#72263836, #72316525

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