#72336797

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 9954 conjugacy classes of subgroups; only the 3033 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,31)(30,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,32)(30,31);(2,4)(3,7)(9,11)(10,14)(12,15)(13,16)(18,20)(22,23)(27,31)(28,29);(2,4)(5,8)(10,15)(12,14)(17,18,21,20)(19,22,24,23)(25,28,30,29)(26,27,32,31);(1,4,6,2)(3,8,7,5)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)(25,29,30,28)(26,31,32,27);(1,5,6,8)(2,7,4,3)(9,12,13,14)(10,11,15,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^10|d^2=h,e^2=g,f^2=gij,b^a=bg,c^a=cj,c^b=ci,d^a=dj,d^c=dh,e^a=ej,e^c=eg,e^d=egh,f^d=fghij,f^e=fh

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336887
maximal normal subgroup of	#72336887

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