#72284196

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|c^2=f,d^2=hi,c^a=ci,c^b=ch,d^a=dhj,d^b=dij,d^c=dg,e^a=efi,e^b=eg,e^d=efj,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285517, #72286185, …
maximal normal subgroup of	#72284709, #72285517, #72286185, …

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