Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,15)(13,14)(17,18,21,23)(19,24,22,20)(25,26,30,28)(27,31,29,32);(2,4)(6,7)(10,13)(11,12)(17,19,21,22)(18,20,23,24)(25,27,30,29)(26,31,28,32);(9,11,14,12)(10,15,13,16)(17,20)(18,22)(19,23)(21,24)(25,27,30,29)(26,32,28,31);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,15,13,16)(17,20)(18,19)(21,24)(22,23)(25,28,30,26)(27,31,29,32);(1,4,5,2)(3,6,8,7)(9,13,14,10)(11,16,12,15)(17,18)(19,24)(20,22)(21,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=fghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2828069);