#72300835

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 9198 conjugacy classes of subgroups; only the 3001 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=hj,b^a=bi,c^b=cg,d^b=dj,d^c=dgh,e^b=eg,e^d=eg,f^a=fhj,f^b=fgh,f^c=fij,f^e=fgh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300929, #72334244
maximal normal subgroup of	#72300929, #72334244

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