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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,10)(7,9,11,12)(14,19)(16,25)(17,21)(20,26)(23,28)(24,27);(2,3)(5,7)(6,9)(8,11)(10,12)(13,14,17,19,18,16,21,25)(15,20,24,26,22,23,27,28);(1,2,4,3)(13,15)(14,20)(16,23)(17,24)(18,22)(19,26)(21,27)(25,28);(1,3,4,2)(7,11)(9,12)(14,16)(15,22)(19,25)(24,27);(13,16,17,25,18,14,21,19)(15,23,24,28,22,20,27,26)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=g,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,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, 7500143);