#72339842

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#2196
commutator	C₄⋊Q₈	C₂³×C₄
Frattini, cosocle	#532	C₂⁴
socle	C₂	C₂×#13047
prime radical	#6348098	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3057 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,e^2=j,f^2=j,g^2=ij,h^2=j,b^a=be,c^a=ch,c^b=cgj,d^a=dj,d^b=dj,d^c=dj,e^a=ej,e^b=eh,e^c=ehj,g^a=gh,g^c=gij,g^e=gij,h^a=hj,h^b=hi,h^c=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339845, #72339852, #72339867
maximal normal subgroup of	#72339845, #72339852, #72339867

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