Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,16,11,15,9,14)(7,12,19,29,20,28,18,27)(8,21,23,32,25,13,24,31)(17,33,22,36,34,26,35,30);(1,2,4,3)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,27)(15,28)(16,29)(17,24)(21,36)(23,34)(25,35)(26,32)(31,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,26)(14,31)(15,21)(16,32)(18,22)(19,35)(20,34)(27,30)(28,33)(29,36);(5,9,11,10)(6,14,15,16)(7,18,20,19)(8,23,25,24)(12,27,28,29)(13,31,21,32)(17,22,34,35)(26,30,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265268);