Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,24,29)(9,14)(10,15)(11,16)(12,19)(13,23,30,25)(18,26,33,36)(20,28)(21,27)(31,32,35,34);(1,2)(3,4)(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,18,23,32,24,33,25,34)(13,26,29,31,30,36,22,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,29)(16,30)(17,31)(19,32)(20,34)(21,33)(27,35)(28,36);(6,14,16,15)(12,27,28,17)(13,22,30,29)(26,35,36,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=efh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284962);