#72236928

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#115133
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	C₂×#23895	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4617 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)(3,11,14,7)(4,15)(5,19)(6,9)(8,18)(10,20)(12,21,28,26)(13,27,29,22)(16,23,31,30)(17,24)(25,32);(1,3)(2,7)(4,16)(5,13)(6,14)(8,23)(9,11)(10,22)(12,17)(15,30)(18,31)(19,27)(20,29)(21,24)(25,26)(28,32);(1,4)(2,8,9,15)(3,12)(5,17)(6,18)(7,21)(10,25,19,24)(11,26)(13,16)(14,28)(20,32)(22,30)(23,27)(29,31)

Polycyclic presentation:

2^10|a^2=ij,c^2=d,b^a=bg,c^a=cf,c^b=ce,d^a=di,d^b=dh,e^a=ei,e^c=eh,f^b=fj,f^c=fi,f^e=fj,g^a=gj,g^c=gij,g^e=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255888, #72255907, #72255953, …
maximal normal subgroup of	#72255888, #72255907, #72255953, …
maximal quotient of	#72236930, #72258218, #72266245

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