#72317741

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, socle	C₂	C₂×#13086
commutator	C₂²×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	#6437452	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=j,d^2=f,g^2=j,h^2=j,c^b=cj,d^a=dg,d^b=df,e^d=eh,f^a=fj,g^a=gh,g^b=gj,g^e=gij,h^a=hi,h^d=hj,h^e=hj,h^g=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317808, #72317889, #72317946, …
maximal normal subgroup of	#72317808, #72317889, #72317946, …

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