#72338313

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	#6353348	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

1862 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)(3,11)(4,16)(5,10)(6,7)(8,15)(9,17)(12,23)(13,25)(14,24)(18,26)(19,28)(20,27)(21,22)(29,30)(31,32);(2,7)(4,17)(8,11)(10,22)(12,30)(18,31)(19,20)(24,25);(3,12)(8,23)(11,29)(13,31)(14,18)(15,30)(24,32)(25,26);(1,3)(2,8)(4,18,19,12)(5,14,21,13)(6,15)(7,11)(9,26,27,23)(10,25,22,24)(16,32,28,29)(17,31,20,30)

Polycyclic presentation:

2^10|d^2=f,g^2=i,h^2=j,b^a=bj,c^a=ci,c^b=ci,d^a=dj,d^c=de,e^a=ej,e^b=ej,e^d=eg,f^c=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	#72338315, #72338330, #72338455
maximal normal subgroup of	#72338315, #72338330, #72338455

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.6);;
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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, 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);