#72302144

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#379
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	C₂×C₂²⋊#95
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=fgh,d^2=i,f^2=j,h^2=j,b^a=bf,c^a=ce,c^b=ceij,d^a=di,d^b=dj,e^a=ei,f^a=fj,f^c=fi,g^c=gi

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302147, #72302152, #72302177
maximal normal subgroup of	#72302147, #72302152, #72302177

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