#72327005

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9609 conjugacy classes of subgroups; only the 3021 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,e^2=hij,b^a=bg,c^a=cj,c^b=cgh,d^a=dhi,e^a=ehi,e^b=ehi,e^c=ej,e^d=ehi,f^a=fj,f^b=fh,f^d=fh,f^e=fgh

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327044
maximal normal subgroup of	#72327044

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