Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419445 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| 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 | |
1319 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)(7,11,17,26)(8,12)(9,20)(10,13)(14,23)(15,29,21,24)(16,25,33,36)(18,30)(19,27)(22,28)(31,35,32,34);(7,15,17,21)(11,24,26,29)(16,32,33,31)(25,35,36,34);(5,7)(6,11,13,26)(8,16)(9,21)(10,17)(12,25,27,36)(14,29,20,24)(15,23)(18,31)(19,33)(22,32)(28,34,30,35);(2,4)(5,8,10,19)(6,12,13,27)(7,16,17,33)(9,22,23,18)(11,25,26,36)(14,30,20,28)(15,31,21,32)(24,34,29,35)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286090, #72286106, #72286224 |
| maximal normal subgroup of | #72286090, #72286106, #72286224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330618);