#72300829

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 9004 conjugacy classes of subgroups; only the 2989 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=gi,c^2=gi,d^2=h,b^a=bg,c^a=cgi,d^a=dh,d^b=dj,d^c=dgij,e^a=egi,e^b=egi,e^c=egi,e^d=egi,f^a=fh,f^b=fghij,f^c=fij,f^e=fghij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300929
maximal normal subgroup of	#72300929

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