Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
847 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)(5,6,11,15)(7,17)(8,22)(9,16,10,14)(12,21)(13,26)(18,34,35,30)(19,28)(20,29)(23,27,36,33)(24,32)(25,31);(5,7,9,19,11,21,10,20)(6,12,16,29,15,17,14,28)(8,23,25,35,24,36,26,18)(13,30,22,33,31,34,32,27);(5,8)(6,13,15,31)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(14,32,16,22)(19,23)(20,36)(21,35)(28,34,29,30);(2,4)(6,14,15,16)(8,24)(12,28,17,29)(13,22,31,32)(18,35)(23,36)(25,26)(27,30,33,34)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280175, #72280664, #72280718, … |
| maximal normal subgroup of | #72280175, #72280664, #72280718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309293);