#72234115

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15257
commutator, Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#6500010	C₂
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

4847 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=ij,b^a=bg,c^a=cf,c^b=ce,d^a=dgh,d^b=dhi,e^a=ei,e^b=ej,e^d=ej,f^d=fij,g^c=gi,h^c=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236976
maximal normal subgroup of	#72236976

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