Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2077 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)(8,10)(9,12)(13,14,18,22)(15,25,26,20)(16,17,23,21)(19,24,27,28);(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,20,22,25)(16,24,23,28)(17,19,21,27);(2,4)(7,12)(9,11)(13,16,27,20,18,23,19,25)(14,21,28,26,22,17,24,15);(6,10)(9,11)(14,22)(16,23)(20,25)(24,28);(13,17,27,15,18,21,19,26)(14,23,28,25,22,16,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717446);