#72236736

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#342
commutator	C₂⁴	C₂²×C₄²
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂²×D₄²	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5496 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=ij,c^2=g,d^2=e,e^2=j,b^a=bij,c^a=chij,c^b=cj,d^b=dj,d^c=df,e^c=eh,f^a=fj,f^c=fi,f^d=fh,f^e=fj,g^d=gi,h^d=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236751, #72263727, #72263733, …
maximal normal subgroup of	#72236751, #72263727, #72263733, …

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