Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478580 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2789 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17155 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8,17,24)(5,11)(6,10)(9,15,27,31)(13,22)(16,21)(18,25,19,29)(20,28)(23,32)(26,30);(1,3)(2,7)(4,16,17,13)(5,15)(6,14)(8,22,24,21)(9,11)(10,12)(18,32,19,30)(20,31)(23,25,26,29)(27,28);(1,4)(2,8,10,24)(3,13)(5,19)(6,17)(7,21,12,22)(9,26,27,23)(11,29,28,25)(14,16)(15,32)(18,20)(30,31);(1,3)(2,9,10,27)(4,13)(5,15)(6,14)(7,11,12,28)(8,23,24,26)(16,17)(18,30)(19,32)(20,31)(21,25,22,29)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385776);