Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35273 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,25,27,28)(16,22,20,21)(17,23,18,26);(1,2)(3,4)(6,9)(10,11)(13,15,17,21,19,27,18,22)(14,20,26,28,24,16,23,25);(2,4)(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,23)(17,25)(18,28)(19,20)(22,24)(26,27);(2,4)(13,16)(14,22)(15,26)(17,25)(18,28)(19,20)(21,24)(23,27);(13,17,19,18)(14,23,24,26)(15,21,27,22)(16,28,20,25)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354855);