Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,15,10,16,11,14)(7,17,20,12,18,28,21,27)(8,22,23,30,24,29,25,13)(19,26,33,35,32,36,34,31);(1,2,3,4)(5,7,9,20,10,18,11,21)(6,12,14,17,16,27,15,28)(8,19,23,33,24,32,25,34)(13,26,29,31,30,36,22,35);(7,18)(19,32)(20,21)(33,34);(5,8,11,25,10,24,9,23)(6,13,15,22,16,30,14,29)(7,19,21,34,18,32,20,33)(12,26,28,35,27,36,17,31)Polycyclic presentation:
2^9|a^2=gh,b^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249844, #72249848, #72249852, … |
| maximal normal subgroup of | #72249844, #72249848, #72249852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322670);