Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2368 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)(6,11)(7,15)(9,24)(13,27)(14,25)(16,26)(17,33)(18,19)(20,30)(21,22)(28,34)(31,35);(1,2)(3,4)(5,6,10,14)(7,16,19,26)(8,20,23,27)(9,25,24,11)(12,18,29,15)(13,22,30,21)(17,34,35,28)(31,32,33,36);(5,7)(6,12)(8,17)(9,18)(10,19)(11,26)(13,28)(14,29)(15,24)(16,25)(20,32)(21,31)(22,33)(23,35)(27,36)(30,34);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,27)(12,28)(14,30)(15,31)(16,32)(18,33)(19,35)(20,25)(21,24)(26,36)(29,34)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,g^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^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324935);