#72224770

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁵
commutator, socle	C₂³	C₂³×C₄²
prime radical	C₂⁵	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=gi,e^2=h,f^2=i,h^2=j,b^a=bg,c^b=ci,d^b=dj,e^a=ej,e^c=ej

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224795, #72224875, #72224917
maximal normal subgroup of	#72224795, #72224875, #72224917

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