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 | C2×#3274 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,20,24)(15,16,26,28)(17,23,19,22)(18,27,25,21);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,25,20,26,17,18)(14,21,23,16,24,27,22,28);(1,2,4,3)(6,9)(10,11)(13,16)(14,15)(17,21)(18,22)(19,27)(20,28)(23,25)(24,26);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,17,20,19)(14,22,24,23)(15,25,26,18)(16,27,28,21);(13,18,19,15,20,25,17,26)(14,16,23,27,24,28,22,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^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,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504216);