Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,21)(13,23)(16,29)(18,27)(20,33)(24,32)(25,30)(26,34)(28,31)(35,36);(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,25,26,29)(13,15,22,27)(19,28,33,35)(20,30)(21,31)(23,32)(34,36);(1,3,4,2)(8,20)(11,24)(12,26)(15,27)(16,30)(19,33)(21,34)(31,36);(5,8,9,21,10,20,23,34)(6,12,13,19,14,26,22,33)(7,16,17,31,18,30,32,36)(11,25,15,28,24,29,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282261);