#72256367

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23900
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3810 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=h,c^a=cg,c^b=ce,d^a=dhi,d^b=dfij,d^c=degh,e^a=eh,e^b=ei,e^d=eh,f^c=fi,g^b=ghj,g^d=gh,g^e=gj,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256380, #72260615
maximal normal subgroup of	#72256380, #72260615

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