Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1286 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)(7,9)(8,22,26,14)(10,18)(11,16)(12,17)(13,20)(15,27)(19,24,34,35)(21,28)(23,30,31,36)(25,32,29,33);(5,7)(6,13)(8,23)(9,18)(10,15)(11,20)(12,21)(14,24)(16,28)(17,27)(19,25)(22,35)(26,31)(29,34)(30,32)(33,36);(5,8)(6,14)(7,19)(9,24)(10,25)(11,29)(12,26)(13,30)(15,31)(16,32)(17,22)(18,33)(20,23)(21,34)(27,36)(28,35);(2,4)(5,9,10,27,12,28,11,13)(6,15,16,7,17,20,18,21)(8,24,25,36,26,35,29,30)(14,31,32,19,22,23,33,34)Polycyclic presentation:
2^9|a^2=gh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358251);