Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419445 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,9,16,11,14)(7,12,19,26,18,27,20,31)(8,21,23,29,22,30,24,13)(17,32,34,28,33,36,35,25);(1,2)(3,4)(5,7,9,18)(6,12,16,27)(8,17,22,33)(10,20,11,19)(13,28,29,25)(14,26,15,31)(21,32,30,36)(23,35,24,34);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,21)(15,30)(16,29)(18,33)(19,35)(20,34)(26,36)(27,28)(31,32);(6,14,16,15)(7,18)(12,26,27,31)(13,21,29,30)(17,33)(19,20)(25,36,28,32)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=ei,f^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,d^b=de,f^b=fi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285557, #72286126, #72286134, … |
| maximal normal subgroup of | #72285557, #72286126, #72286134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327034);