Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#396 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1342 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)(5,6,11,14,10,15,9,16)(7,17,20,27,21,31,19,12)(8,13,23,32,24,30,25,29)(18,33,22,26,34,28,35,36);(5,7,9,19,10,21,11,20)(6,12,14,17,15,27,16,31)(8,22,23,34,24,35,25,18)(13,28,29,26,30,33,32,36);(5,8,9,23,10,24,11,25)(6,13,14,29,15,30,16,32)(7,18,19,22,21,34,20,35)(12,26,17,33,27,36,31,28);(2,4)Polycyclic presentation:
2^9|a^2=efi,b^2=e,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303141);