Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,12)(7,17,18,13,21,28,20,29)(8,22,25,30,23,26,24,14)(19,27,32,35,34,36,33,31);(1,2)(3,4)(6,12,16,15)(7,18,21,20)(8,23)(14,22,30,26)(19,33,34,32)(24,25)(27,36)(31,35);(2,4)(5,7)(6,13)(8,19)(9,20)(10,18)(11,21)(12,17)(14,27)(15,28)(16,29)(22,35)(23,34)(24,33)(25,32)(26,31)(30,36);(5,8,9,24,11,23,10,25)(6,14,15,22,16,30,12,26)(7,19,20,33,21,34,18,32)(13,27,28,35,29,36,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272451);