Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437445 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62814 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22);(1,3,5,4)(2,6,8,7)(10,15)(12,13)(16,23)(17,18)(19,21)(22,24);(1,3,5,4)(2,7,8,6)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72278796, #72278872, #72279044, … |
| maximal normal subgroup of | #72278796, #72278872, #72279044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360135);