Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| prime radical | #44913 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(14,18)(16,21)(19,24)(20,25)(23,26)(27,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,14,16,18,17,20,21,25)(15,19,23,24,22,27,26,28);(6,10)(7,11)(13,15,17,22)(14,19,20,27)(16,23,21,26)(18,24,25,28);(2,4)(6,10)(9,12)(14,20)(15,22)(18,25)(23,26);(13,14,21,25,17,20,16,18)(15,19,26,28,22,27,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434751);