Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6379185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,12,5,20)(3,13,17,8,16,26,21,25)(4,10,18,27,19,9,22,28)(14,30,15,23,31,24,32,29);(1,3)(2,8)(4,14)(5,21)(6,17)(7,16)(9,23)(10,29)(11,26)(12,25)(13,20)(15,18)(19,31)(22,32)(24,28)(27,30);(1,4)(2,9)(3,14)(5,18)(6,22)(7,19)(8,23)(10,12)(11,27)(13,24)(15,21)(16,31)(17,32)(20,28)(25,29)(26,30);(1,4)(2,10)(3,15)(5,22)(6,18)(7,19)(8,24)(9,12)(11,27)(13,23)(14,21)(16,32)(17,31)(20,28)(25,30)(26,29);(3,16)(8,25)(13,26)(14,31)(15,32)(17,21)(23,29)(24,30)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,h^c=hi,f^d=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72215784, #72215938, #72219686, … |
| maximal normal subgroup of | #72215784, #72215938, #72219686, … |
| maximal quotient of | #72312215, C2×#6379191 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286387);