#72280875

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1438
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
socle	C₂²	#7555
prime radical	C₂⁴×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

4897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=degj,d^2=i,e^2=j,b^a=bfh,c^a=cd,c^b=ce,d^a=dj,e^a=ei,e^b=ej,f^c=fij,g^a=gj,h^c=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280953, #72302152, #72302198
maximal normal subgroup of	#72280953, #72302152, #72302198

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