Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,14)(7,15,20,17)(8,21,24,31)(9,25,27,13)(11,19,29,18)(12,26,30,23)(16,33,22,28)(32,35,34,36);(5,7)(6,11)(8,22)(9,19)(10,20)(12,28)(13,17)(14,29)(15,25)(16,24)(18,27)(21,32)(23,35)(26,36)(30,33)(31,34);(5,8,10,24)(6,12,14,30)(7,16)(9,26,27,23)(11,28)(13,31,25,21)(15,32)(17,34)(18,36)(19,35)(20,22)(29,33);(5,6)(7,17,20,15)(8,12)(9,13)(10,14)(11,19,29,18)(16,34,22,32)(21,23)(24,30)(25,27)(26,31)(28,35,33,36)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292729);