Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,11,16)(7,17,19,22,21,32,20,13)(9,14,25,30,27,23,26,31)(18,29,24,35,33,36,34,28);(5,7,11,19,12,21,10,20)(6,13,16,17,8,22,15,32)(9,24,26,33,27,34,25,18)(14,29,31,35,23,36,30,28);(5,8,12,6)(7,17)(9,23,27,14)(10,15,11,16)(13,19)(18,29)(20,22)(21,32)(24,28)(25,30,26,31)(33,36)(34,35);(2,4)(5,9)(6,14)(7,18,21,33)(8,23)(10,25)(11,26)(12,27)(13,28,22,35)(15,30)(16,31)(17,29,32,36)(19,24,20,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=efi,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339194);