#72227588

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#6342371
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

215783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|d^2=j,f^2=j,b^a=bj,c^a=cg,c^b=cf,d^a=dj,d^b=dj,d^c=dh,e^c=ehi,e^d=ej,f^a=fj,f^b=fj,f^c=fj,f^d=fj,f^e=fj,g^b=gj,g^d=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227590, #72227629, #72227648, …
maximal normal subgroup of	#72227590, #72234499, #72241085, …
maximal quotient of	#72234942, #72235015, #72268010, …

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