#72328577

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 9948 conjugacy classes of subgroups; only the 3113 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:

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

Polycyclic presentation:

2^10|e^2=gh,f^2=ij,b^a=bi,c^b=cg,d^b=dh,e^a=eij,e^b=egh,e^d=eh,f^a=fij,f^b=fh,f^d=fij,f^e=fgi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328746
maximal normal subgroup of	#72328746
maximal quotient of	#72331147

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