Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,17)(15,22,25,28)(16,26,21,18)(20,27,23,24);(1,2)(3,4)(13,15,19,25)(14,20,17,23)(16,22,21,28)(18,27,26,24);(2,4)(6,9)(10,12)(13,16)(14,18)(15,22)(17,26)(19,21)(20,27)(23,24)(25,28);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,23,25,20)(16,26,21,18)(22,24,28,27);(14,17)(16,21)(20,23)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355928);