Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417566 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1954 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)(7,15,18,11)(8,12)(9,21)(10,13)(14,24)(16,25,22,29)(17,31,34,26)(19,30)(20,27)(23,28)(32,35,33,36);(7,16,18,22)(11,25,15,29)(17,33,34,32)(26,36,31,35);(5,7)(6,11)(8,17)(9,22)(10,18)(12,26)(13,15)(14,29)(16,24)(19,32)(20,34)(21,25)(23,33)(27,31)(28,35)(30,36);(2,4)(5,8,10,20)(6,12,13,27)(7,17,18,34)(9,23,24,19)(11,26,15,31)(14,30,21,28)(16,32,22,33)(25,35,29,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325157);