Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2410 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)(17,18,22,23)(19,24,27,31)(20,28,29,21)(25,26,32,30);(5,6,7,8)(10,11)(14,15)(18,23)(20,29)(24,31)(25,32);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19)(18,24)(20,25)(21,30)(22,27)(23,31)(26,28)(29,32);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,20,22,29)(18,21,23,28)(19,25,27,32)(24,30,31,26);(9,12)(10,14)(11,15)(13,16)(17,21)(18,20)(19,26)(22,28)(23,29)(24,32)(25,31)(27,30)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3372138);