Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1329 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)(5,6,9,15,11,16,10,13)(7,17,18,29,21,28,20,12)(8,22,24,30,25,14,26,32)(19,33,23,27,35,31,34,36);(5,7)(6,12)(8,23)(9,20)(10,18)(11,21)(13,17)(14,31)(15,28)(16,29)(19,24)(22,33)(25,34)(26,35)(27,32)(30,36);(6,13)(7,18)(9,10)(14,30)(15,16)(17,28)(19,23)(20,21)(22,32)(24,26)(31,33)(34,35);(2,4)(5,8,11,25)(6,14,16,22)(7,19)(9,24,10,26)(12,27)(13,30,15,32)(17,31)(18,23)(20,34)(21,35)(28,33)(29,36)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,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, 335186);