Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2510 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)(10,11)(12,16)(17,18)(19,23)(20,22)(21,28)(24,29)(25,27)(26,32)(30,31);(5,6,7,8)(10,11)(14,15)(18,20)(19,25)(24,28)(26,31);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,25,27)(21,24,29,28)(26,32,31,30);(9,12)(10,14)(11,15)(13,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)Polycyclic presentation:
2^9|b^2=fgh,c^2=fg,d^2=fg,e^2=f,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, 2668491);