#72262471

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#213828
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6499870	C₂
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3077 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:

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

Polycyclic presentation:

2^10|a^2=e,e^2=j,c^a=cg,c^b=cf,d^a=dfij,d^b=dhj,d^c=dj,e^c=ei,e^d=eh,f^a=fh,f^e=fj,g^a=gi,g^b=gh,h^a=hj,i^b=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262472, #72262492, #72262493, …
maximal normal subgroup of	#72262472, #72262492, #72262493, …
maximal quotient of	#72338771, #72340460, #72340520, …

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