Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,15,10,24)(7,16,19,26,18,28,23,12)(8,20,22,13,21,30,25,29)(17,32,34,36,33,27,35,31);(1,2)(3,4)(8,21)(12,26)(13,29)(16,28)(20,30)(22,25)(27,32)(31,36);(2,4)(5,7,9,23,11,18,10,19)(6,12,14,28,15,26,24,16)(8,17,22,35,21,33,25,34)(13,27,30,36,29,32,20,31);(5,8)(6,13)(7,17)(9,22)(10,25)(11,21)(12,27)(14,30)(15,29)(16,31)(18,33)(19,34)(20,24)(23,35)(26,32)(28,36)Polycyclic presentation:
2^9|a^2=f,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285577, #72285592, #72285631 |
| maximal normal subgroup of | #72285577, #72285592, #72285631 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332206);