Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
929 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)(5,6,11,14)(7,15,20,27)(8,21,26,23)(9,10,28,29)(12,19,17,18)(13,24,32,25)(16,30,35,33)(22,36,34,31);(5,7,11,20)(6,12,14,17)(8,22)(9,27,28,15)(10,19,29,18)(13,31)(16,24)(21,33)(23,30)(25,35)(26,34)(32,36);(5,8)(6,13)(7,16)(9,23)(10,24)(11,26)(12,30)(14,32)(15,31)(17,33)(18,22)(19,34)(20,35)(21,28)(25,29)(27,36);(2,4)(5,9,10,14,11,28,29,6)(7,17,18,27,20,12,19,15)(8,23,24,32,26,21,25,13)(16,33,22,36,35,30,34,31)Polycyclic presentation:
2^9|a^2=i,b^2=ef,d^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,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328642);