Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2433 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)(11,13)(12,14)(15,16)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31);(5,6,7,8)(10,12)(11,15)(19,26)(20,25)(27,32)(28,31);(5,6)(7,8)(9,11,14,15)(10,13,12,16)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32);(1,2)(3,4)(9,12,14,10)(11,13,15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(9,13)(10,15)(11,12)(14,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668518);