Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60425 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,17)(7,18)(8,23)(9,26,28,31)(10,16,11,29)(13,21)(14,24)(15,30,36,27)(19,35)(20,33)(22,32)(25,34);(1,2)(3,4)(5,7,10,21,12,22,11,20)(6,13,16,32,17,33,29,18)(8,9,19,27,25,28,24,30)(14,31,23,36,35,26,34,15);(2,4)(5,8,10,24,12,25,11,19)(6,14,16,34,17,35,29,23)(7,9,20,27,22,28,21,30)(13,31,18,36,33,26,32,15);(5,9)(6,15)(7,19)(8,20)(10,27)(11,30)(12,28)(13,14)(16,31)(17,36)(18,34)(21,25)(22,24)(23,32)(26,29)(33,35)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304380);