Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42890 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,17)(16,25)(18,21)(19,20)(22,23)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,22,24,27)(16,20,19,25)(17,26,28,23);(1,2,3,4)(6,9)(10,11)(13,15,18,24)(14,17,21,28)(16,23,19,26)(20,27,25,22);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,20,21,25)(15,23,24,26)(17,27,28,22);(13,17,18,28)(14,15,21,24)(16,22,19,27)(20,26,25,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592217);