Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62819 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,21)(4,15)(8,23,26,31)(9,16,11,22)(13,27)(17,29)(18,30)(20,24,32,25);(1,2)(3,8)(4,16)(5,11)(6,9)(7,18)(10,19)(12,24)(13,20)(14,25)(15,21)(17,23)(22,30)(26,28)(27,31)(29,32);(1,3,6,14)(2,8,9,25)(4,17,18,13)(5,12,19,28)(7,20,16,23)(10,26,11,24)(15,29,30,27)(21,32,22,31);(3,12)(8,24)(13,27)(14,28)(17,29)(20,31)(23,32)(25,26)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260645, #72260669 |
| maximal normal subgroup of | #72260645, #72260669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94017);