Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2510 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)(7,17)(8,13,24,32)(9,16)(10,14)(11,15)(12,20)(18,29,34,33)(19,28)(21,27)(22,26,35,36)(23,31,25,30);(1,2)(3,4)(5,7)(6,12)(8,22)(9,21)(10,20)(11,19)(13,29)(14,28)(15,27)(16,17)(18,25)(23,34)(24,35)(26,31)(30,36)(32,33);(2,4)(5,8,9,23,11,24,10,25)(6,13,14,30,15,32,16,31)(7,18,20,35,19,34,21,22)(12,26,17,33,27,36,28,29);(7,19)(12,27)(17,28)(18,34)(20,21)(22,35)(26,36)(29,33)Polycyclic presentation:
2^9|a^2=h,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265165);