Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417704 |
| 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 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2214 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)(5,6,9,14,11,16,10,15)(7,17,18,28,20,26,21,12)(8,22,23,13,25,29,24,30)(19,33,34,31,35,27,32,36);(7,18,20,21)(12,26,28,17)(19,34,35,32)(27,31,33,36);(2,4)(5,7)(6,12)(8,19)(9,21)(10,18)(11,20)(13,27)(14,26)(15,17)(16,28)(22,36)(23,32)(24,34)(25,35)(29,31)(30,33);(5,8,11,25)(6,13,16,30)(7,19,20,35)(9,24,10,23)(12,27,28,33)(14,22,15,29)(17,31,26,36)(18,32,21,34)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 326254);