Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,10,16,9,15)(7,17,19,28,21,27,20,12)(8,13,25,31,24,32,23,30)(18,33,35,26,34,29,22,36);(1,2)(3,4)(5,7,9,20,10,21,11,19)(6,12,14,17,16,28,15,27)(8,22,23,34,24,35,25,18)(13,29,31,36,32,33,30,26);(2,4)(5,8)(6,13)(7,18)(9,25)(10,24)(11,23)(12,26)(14,30)(15,31)(16,32)(17,33)(19,22)(20,35)(21,34)(27,29)(28,36);(6,14)(7,19)(9,11)(13,30)(15,16)(17,27)(18,22)(20,21)(23,25)(29,33)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,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, 326376);