Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,14,16)(11,12,15,13)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32);(6,8)(9,11,14,15)(10,12,16,13)(18,23)(19,24)(21,30)(28,32);(5,6,7,8)(9,12,14,13)(10,11,16,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31);(1,2)(3,4)(9,10)(11,12)(13,15)(14,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32);(9,13,14,12)(10,15,16,11)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 992436);