Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16)(7,18,21,13)(8,23)(9,17,11,15)(12,22,29,20)(14,25)(19,28)(24,33)(26,32)(27,31)(30,34)(35,36);(1,2,3,4)(5,7,9,20,10,21,11,22)(6,12,15,18,16,29,17,13)(8,24,25,35,27,34,26,19)(14,30,23,36,32,33,31,28);(5,7,10,21)(6,13,16,18)(8,19,27,35)(9,22,11,20)(12,15,29,17)(14,30,32,33)(23,28,31,36)(24,25,34,26);(2,4)(5,8)(6,14)(7,19)(9,26)(10,27)(11,25)(12,28)(13,30)(15,31)(16,32)(17,23)(18,33)(20,34)(21,35)(22,24)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326173);