Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2487 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)(7,18)(8,21,24,31)(9,13)(10,26)(11,16)(12,17)(14,23,32,27)(15,20)(19,34,36,28)(22,33,35,30)(25,29);(1,2)(3,4)(5,7)(6,13)(8,19)(9,11)(10,25)(12,20)(14,28)(15,16)(17,29)(18,26)(21,33)(22,27)(23,35)(24,36)(30,31)(32,34);(2,4)(5,8,11,27,12,24,10,23)(6,14,16,31,17,32,26,21)(7,19,9,22,20,36,25,35)(13,28,15,30,29,34,18,33);(5,9)(6,15)(7,10)(8,22)(11,20)(12,25)(13,26)(14,30)(16,29)(17,18)(19,23)(21,28)(24,35)(27,36)(31,34)(32,33)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281305);