Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62876 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3485 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,9)(5,10,18,7)(6,11)(12,23)(13,22)(14,21,26,19)(15,28)(16,24,30,20)(17,25)(27,32,29,31);(2,7)(3,12)(5,18)(8,21)(9,20)(10,11)(13,15)(16,30)(19,23)(22,32)(24,25)(28,31);(1,3,6,12)(2,8,11,23)(4,15,17,13)(5,14,18,26)(7,19,10,21)(9,22,25,28)(16,29,30,27)(20,31,24,32);(1,4)(2,9)(3,13)(5,16)(6,17)(7,20)(8,22)(10,24)(11,25)(12,15)(14,27)(18,30)(19,31)(21,32)(23,28)(26,29)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72240648, #72249653, #72261759 |
| maximal normal subgroup of | #72240648, #72249653, #72261759 |
| maximal quotient of | #72255738, #72255885, #72262246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400680);