Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,16)(15,20,17,21)(18,22,28,26)(23,24,27,25);(6,9)(10,12)(13,15,19,17)(14,20,16,21)(18,24,28,25)(22,27,26,23);(2,4)(13,16,19,14)(15,23,17,27)(18,22,28,26)(20,24,21,25);(5,7,8,11)(6,10,9,12)(13,17,19,15)(14,21,16,20)(18,25,28,24)(22,23,26,27);(15,24)(17,25)(18,28)(20,27)(21,23)(22,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482460);