Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
15968 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)(5,6,10,14,11,15,9,22)(7,16,18,12,19,27,23,26)(8,13,20,29,21,30,24,28)(17,32,35,36,34,31,33,25);(5,7)(6,12)(8,17)(9,23)(10,18)(11,19)(13,25)(14,27)(15,26)(16,22)(20,35)(21,34)(24,33)(28,31)(29,32)(30,36);(5,8)(6,13)(7,17)(9,24)(10,20)(11,21)(12,25)(14,29)(15,30)(16,31)(18,35)(19,34)(22,28)(23,33)(26,36)(27,32);(2,4)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257114, #72257145, #72257147, … |
| maximal normal subgroup of | #72257114, #72257145, #72257147, … |
| maximal quotient of | C2×#392131 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299327);