Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #35301 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,21)(20,24,30,32)(22,25)(23,26)(27,31,28,29);(1,3,6,11)(2,7)(4,12,5,13)(8,10)(9,15)(14,16)(17,19,21,18,23,25,22,26)(20,28,29,32,30,27,31,24);(3,11)(7,15)(10,16)(12,13)(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,28)(26,32);(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,28,32,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318610, #72318819, #72318856 |
| maximal normal subgroup of | #72318610, #72318819, #72318856 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108830);