Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 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)(13,14,20,17,19,24,18,23)(15,16,25,21,26,28,22,27);(2,4)(6,9)(10,11)(13,15)(14,21)(16,17)(18,25)(19,26)(20,22)(23,28)(24,27);(5,7,8,12)(6,10,9,11)(13,16,20,21,19,28,18,27)(14,22,17,15,24,25,23,26);(13,17,20,24,19,23,18,14)(15,21,25,28,26,27,22,16);(13,18,19,20)(14,23,24,17)(15,25,26,22)(16,21,28,27)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373397);