Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | #60415 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14,11,16,9,15)(7,12,19,27,20,28,18,31)(8,21,25,30,23,32,24,13)(17,33,35,36,34,29,22,26);(1,2,4,3)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,27)(15,31)(16,28)(17,25)(21,26)(23,35)(24,34)(30,33)(32,36);(5,8)(6,13,16,30)(7,17)(9,24)(10,25)(11,23)(12,26,28,36)(14,21,15,32)(18,22)(19,35)(20,34)(27,33,31,29);(1,3,4,2)(8,23)(13,30)(17,34)(21,32)(22,35)(24,25)(26,36)(29,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281562);