Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2206 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)(7,16)(8,20)(9,11)(10,15)(12,17)(13,22)(14,25)(18,27)(19,28)(21,33)(23,32)(24,31)(26,29)(30,34)(35,36);(6,11)(7,17)(9,25)(13,20)(14,15)(16,28)(18,34)(19,26)(21,35)(22,23)(30,33)(31,32);(5,7,10,19)(6,12,15,29)(8,21,24,34)(9,26,25,17)(11,16,14,28)(13,30,32,33)(18,23,35,22)(20,27,31,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,20)(12,27)(14,31)(15,32)(16,33)(17,34)(19,35)(21,26)(22,25)(28,30)(29,36)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361307);