Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,9,15,10,16,27,14)(7,17,19,11,21,29,20,13)(8,22,25,32,26,33,24,12)(18,31,35,30,36,28,23,34);(5,7,9,19,10,21,27,20)(6,11,14,17,16,13,15,29)(8,23,24,36,26,35,25,18)(12,30,22,28,32,34,33,31);(2,4)(5,8)(6,12)(7,18)(9,24)(10,26)(11,28)(13,31)(14,22)(15,33)(16,32)(17,34)(19,23)(20,35)(21,36)(25,27)(29,30);(5,7)(6,13)(8,18)(9,20)(10,21)(11,16)(12,31)(14,17)(15,29)(19,27)(22,34)(23,25)(24,35)(26,36)(28,32)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326208);