Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16,10,15,12,27)(7,17,20,32,19,13,21,30)(8,22,24,31,9,25,26,14)(18,33,34,28,23,36,35,29);(1,2,4,3)(5,7,12,21,10,19,11,20)(6,13,27,32,15,17,16,30)(8,23,26,34,9,18,24,35)(14,29,25,36,31,28,22,33);(5,8,10,9)(6,14,15,31)(7,18,19,23)(11,26,12,24)(13,28,17,29)(16,25,27,22)(20,35,21,34)(30,33,32,36);(1,3,4,2)(5,9,10,8)(6,14,15,31)(7,18,19,23)(11,26,12,24)(13,29,17,28)(16,22,27,25)(20,34,21,35)(30,33,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304669);