Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460330 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 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)(5,6,10,15,11,16,9,13)(7,17,20,26,21,28,18,12)(8,22,24,30,25,29,23,14)(19,33,35,36,34,27,32,31);(5,7,9,18,11,21,10,20)(6,12,15,17,16,26,13,28)(8,19,23,32,25,34,24,35)(14,27,22,31,30,33,29,36);(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(19,34)(20,21)(22,29)(24,25)(27,36)(31,33);(2,4)(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,13,29)(7,19,18,32,21,34,20,35)(12,27,17,31,26,33,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^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,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367526);