Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
1877 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)(7,17,19,12,20,27,25,28)(8,13,22,31,24,30,23,32)(18,33,21,26,35,29,34,36);(1,2)(3,4)(5,7)(6,12)(8,21)(9,25)(10,20)(11,19)(13,29)(14,27)(15,17)(16,28)(18,23)(22,35)(24,34)(26,32)(30,33)(31,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,24)(11,23)(12,26)(14,32)(15,31)(16,30)(17,29)(19,34)(20,35)(21,25)(27,33)(28,36);(6,14)(8,22)(9,11)(12,27)(13,30)(15,16)(17,28)(18,34)(19,25)(21,35)(23,24)(29,33)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361860);