#72310350

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#213828
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6499875	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2627 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)(4,15)(5,9)(6,10)(7,14)(8,16)(12,21)(13,20)(17,28)(18,23)(19,26)(22,29)(24,27)(25,30)(31,32);(3,12)(4,16)(7,19,11,21)(8,15)(9,23)(13,25,27,32)(14,26)(20,30)(22,28)(24,31);(1,3,5,12)(2,7,9,21)(4,13,17,25)(6,14,18,26)(8,20,22,31)(10,11,23,19)(15,24,28,30)(16,27,29,32);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,22)(10,15)(11,24)(12,25)(14,27)(18,29)(19,30)(21,31)(23,28)(26,32)

Polycyclic presentation:

2^10|b^2=e,c^2=f,b^a=bg,c^a=cg,c^b=cef,d^a=dj,d^b=dgij,e^a=ei,e^c=eh,e^d=ei,f^a=fi,f^b=fh,g^b=gi,g^c=gi,g^f=gj,h^a=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310354, #72314648, #72314682
maximal normal subgroup of	#72310354, #72314648, #72314682

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