Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,10)(9,24)(11,15)(12,16)(13,20)(14,26)(18,21)(19,28)(22,23)(25,30)(27,29)(31,34)(32,35)(33,36);(5,7,10,20,12,22,11,21)(6,13,15,17,16,18,8,23)(9,19,25,33,27,35,26,34)(14,28,29,31,30,32,24,36);(2,4)(5,8,10,6,12,15,11,16)(7,18,20,23,22,13,21,17)(9,24,25,14,27,29,26,30)(19,32,33,36,35,28,34,31);(5,9)(6,14)(7,19)(8,24)(10,25)(11,26)(12,27)(13,28)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(22,35)(23,36)Polycyclic presentation:
2^9|b^2=e,c^2=e,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,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302019);