#72326751

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 9148 conjugacy classes of subgroups; only the 2997 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:

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

Polycyclic presentation:

2^10|c^2=j,d^2=i,e^2=gh,b^a=bg,c^b=cgh,d^b=di,d^c=dij,e^b=ei,e^c=eghj,e^d=eghi,f^a=fj,f^b=fhij,f^c=fj,f^d=fhij,f^e=fi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326780
maximal normal subgroup of	#72326780

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