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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2075 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)(15,20)(16,25)(17,24)(18,22)(19,23)(21,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,17,23,20)(16,21,26,25)(18,27,28,22);(1,2,3,4)(6,9)(10,11)(13,16)(14,21)(15,18)(17,22)(19,26)(20,27)(23,25)(24,28);(5,7,8,12)(6,10,9,11)(13,17,19,20)(14,15,23,24)(16,22,26,27)(18,25,28,21);(13,18)(14,22)(15,16)(17,21)(19,28)(20,25)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,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, 6592216);