#72338304

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13561
commutator	#127	C₂³×C₄
Frattini, cosocle	#493	C₂⁴
prime radical	#6353347	C₂
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ C₄×#41 ◁ #60646 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2034 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:

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

Polycyclic presentation:

2^10|d^2=e,g^2=i,h^2=j,b^a=bj,c^b=cj,d^a=dij,d^b=dij,d^c=df,e^a=ej,e^b=ej,e^c=eg,f^d=fgi,f^e=fhij,g^c=gi,g^d=gh,g^e=gi,g^f=gi,h^c=hi,h^d=hij,h^e=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338309, #72338315, #72338319, …
maximal normal subgroup of	#72338309, #72338315, #72338319, …

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