Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1131 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)(7,13,21,30)(8,14)(9,26)(10,18)(11,17)(12,16)(15,24)(19,29,35,28)(20,36,23,32)(22,34,27,31)(25,33);(5,7)(6,13,16,30)(8,23)(9,20)(10,27)(11,22)(12,21)(14,32,26,36)(15,29,33,28)(17,34,18,31)(19,24)(25,35);(2,4)(5,8,12,9)(6,14,16,26)(7,19,21,35)(10,25,11,24)(13,28,30,29)(15,18,33,17)(20,27,23,22)(31,36,34,32);(5,9,12,8)(6,15,16,33)(7,20,21,23)(10,24,11,25)(13,29,30,28)(14,17,26,18)(19,22,35,27)(31,32,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379089);