Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2371 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)(7,13)(8,21)(9,16)(10,15)(11,17)(12,19)(14,25)(18,27)(20,30)(22,29)(23,33)(24,32)(26,28)(31,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,16)(14,31)(15,30)(17,28)(18,23)(21,36)(24,35)(25,34)(27,32)(29,33);(2,4)(5,7)(6,13)(8,18)(9,19)(10,26)(11,20)(12,16)(14,29)(15,28)(17,30)(21,36)(22,23)(24,34)(25,35)(27,32)(31,33);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,27)(13,29)(15,21)(16,32)(17,33)(19,35)(20,34)(22,26)(28,36)(30,31)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324861);