Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32976 |
| prime radical | #3481 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,4,16)(3,12,19,23,13,28,14,8)(5,9,20,24,18,25,17,31)(15,27,32,21,29,26,30,22);(1,3,4,14,6,13,7,19)(2,8,10,12,11,23,16,28)(5,15,17,30,18,29,20,32)(9,22,24,27,25,21,31,26);(2,9)(3,13)(8,21)(10,24)(11,25)(12,26)(14,19)(15,29)(16,31)(22,23)(27,28)(30,32);(5,18)(15,29)(17,20)(30,32)Polycyclic presentation:
2^9|a^2=eh,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231700, #72231703, … |
| maximal normal subgroup of | #72231596, #72231700, #72231703, … |
| maximal quotient of | #72250688, #72280127, C2×#512046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419242);