Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1878 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)(5,6,12,17)(7,18,21,27)(8,22,25,31)(9,20,13,28)(10,16,11,15)(14,26,30,24)(19,23,35,29)(32,33,36,34);(1,2,3,4)(5,7,10,20,12,21,11,28)(6,13,15,18,17,9,16,27)(8,19,24,33,25,35,26,34)(14,29,22,36,30,23,31,32);(5,8,11,24,12,25,10,26)(6,14,16,22,17,30,15,31)(7,19,20,34,21,35,28,33)(9,23,27,36,13,29,18,32);(2,4)(5,9,12,13)(6,7,17,21)(8,23,25,29)(10,18,11,27)(14,19,30,35)(15,28,16,20)(22,34,31,33)(24,36,26,32)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330142);