Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21452 |
| prime radical | #33530 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,13,23,12)(3,14,18,29,17,30,22,27)(4,9,5,20,19,10,21,28)(8,15,25,16,24,31,26,32);(1,3)(2,8)(4,16)(5,15)(6,22)(7,17)(9,27)(10,29)(11,26)(12,25)(13,24)(14,28)(18,23)(19,32)(20,30)(21,31);(4,19)(5,21)(9,10)(14,30)(15,31)(16,32)(20,28)(27,29);(1,4,6,5,7,19,23,21)(2,9,11,20,13,10,12,28)(3,15,18,16,17,31,22,32)(8,14,25,29,24,30,26,27)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, C2×#495736, #72217918, … |
| maximal normal subgroup of | #72213945, C2×#495736, #72217918, … |
| maximal quotient of | #72216636, #72216647, #72216650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402932);