Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,15,11,16,10,14)(7,17,21,19,22,29,20,12)(8,23,26,13,27,31,25,32)(18,34,35,30,36,28,24,33);(1,2,4,3)(5,7,9,21,11,22,10,20)(6,12,15,17,16,19,14,29)(8,24,25,36,27,35,26,18)(13,30,23,34,32,33,31,28);(5,8)(6,13)(7,18)(9,26)(10,25)(11,27)(12,28)(14,23)(15,31)(16,32)(17,33)(19,34)(20,24)(21,35)(22,36)(29,30);(1,3,4,2)(5,6)(7,19)(8,13)(9,15)(10,14)(11,16)(12,22)(17,20)(18,34)(21,29)(23,25)(24,33)(26,31)(27,32)(28,36)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269860);