Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,26,10,16,9,15)(7,13,24,12,19,30,18,28)(8,20,22,32,23,14,25,31)(17,29,21,27,33,36,34,35);(1,2)(3,4)(5,7)(6,12)(8,17)(9,24)(10,19)(11,18)(13,26)(14,27)(15,30)(16,28)(20,35)(21,25)(22,34)(23,33)(29,31)(32,36);(2,4)(5,7)(6,13)(8,21,23,34)(9,18)(10,19)(11,24)(12,26)(14,27,20,35)(15,28)(16,30)(17,22,33,25)(29,31,36,32);(5,8)(6,14)(7,17)(9,25)(10,23)(11,22)(12,27)(13,29)(15,32)(16,20)(18,34)(19,33)(21,24)(26,31)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=efi,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364803);