Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,22,19,15,18,21)(16,20,25,24,27,23,26,28);(2,4)(6,9)(10,11)(13,15)(14,19)(16,24)(17,22)(18,21)(20,25)(23,26)(27,28);(5,7,8,12)(6,10,9,11)(13,16,17,25,19,27,18,26)(14,20,22,24,15,23,21,28);(13,16,17,25,19,27,18,26)(14,20,22,24,15,23,21,28);(13,17,19,18)(14,21,15,22)(16,25,27,26)(20,28,23,24)Polycyclic presentation:
2^9|a^2=fg,c^2=ghi,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373381);