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 | #25416 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14,19,21)(15,22,17,18)(16,24,26,20)(23,27,25,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,18,14,19,17,22,21)(16,25,27,24,26,23,28,20);(5,8,9,12)(6,10,7,11)(13,16,19,26)(14,20,21,24)(15,23,17,25)(18,27,22,28);(2,4)(6,7)(10,11)(14,21)(15,17)(16,26)(27,28);(13,17,18,21,19,15,22,14)(16,25,27,24,26,23,28,20)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7453054);