Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,14,26)(8,20)(9,10)(13,28,30,25)(15,32,23,27)(16,33,31,36)(17,29)(18,24)(21,22)(34,35);(5,6,8,12,10,15,20,27,11,14,22,26,9,23,21,32)(7,16,17,33,18,13,29,25,19,31,34,36,24,30,35,28);(2,4)(5,7)(6,13)(8,17)(9,24)(10,18)(11,19)(12,25)(14,30)(15,31)(16,23)(20,29)(21,35)(22,34)(26,28)(27,36)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249869, #72249872, #72249885, … |
| maximal normal subgroup of | #72249869, #72249872, #72249885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55490);