Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 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)(7,10)(8,11)(9,12)(13,14,19,17)(15,23,25,28)(16,22,18,21)(20,26,27,24);(7,8)(10,11)(13,15,16,26,19,25,18,24)(14,20,21,28,17,27,22,23);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,18)(14,21,17,22)(15,24,25,26)(20,23,27,28);(5,8,9,7)(6,11,12,10)(13,17,19,14)(15,20,25,27)(16,22,18,21)(23,26,28,24);(6,12)(10,11)(14,17)(20,27)(21,22)(23,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6395035);