Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24236 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1025 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)(3,16)(4,21)(5,13)(6,26)(7,28)(8,15)(9,18)(10,22)(11,24)(12,27)(14,25)(17,29)(19,30)(20,23)(31,32);(1,3,6,18,8,21,27,20)(2,9,12,16,15,23,26,4)(5,17,24,31,25,19,7,10)(11,30,13,32,28,29,14,22);(1,4)(2,10)(3,11)(5,22)(6,9)(7,29)(8,16)(12,17)(13,18)(14,20)(15,31)(19,26)(21,28)(23,27)(24,30)(25,32);(1,5,6,24,8,25,27,7)(2,11,12,13,15,28,26,14)(3,17,18,31,21,19,20,10)(4,22,9,30,16,32,23,29)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407934);