Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,13)(7,12,19,27,20,25,17,28)(8,21,23,30,22,29,24,14)(18,32,31,26,34,35,33,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,17)(11,20)(13,28)(14,26)(15,27)(16,25)(21,35)(22,34)(23,31)(24,33)(29,32)(30,36);(2,4)(6,13)(7,17)(8,22)(9,10)(12,25)(14,29)(15,16)(18,33)(19,20)(21,30)(31,34)(32,35);(5,8,9,24,11,22,10,23)(6,14,15,29,16,30,13,21)(7,18,19,33,20,34,17,31)(12,26,27,32,25,36,28,35)Polycyclic presentation:
2^9|a^2=f,b^2=egh,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380330);