Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15836 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
869 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)(3,10,17,28)(4,18)(5,11)(6,12)(7,13)(8,15,24,16)(9,21)(14,29,31,23)(19,32,30,25)(20,27)(22,26);(1,3,6,16,7,17,5,15)(2,8,11,10,13,24,12,28)(4,19,21,14,22,30,20,31)(9,25,18,29,27,32,26,23);(1,4,6,21,7,22,5,20)(2,9,11,18,13,27,12,26)(3,14,15,19,17,31,16,30)(8,23,28,32,24,29,10,25);(1,3,7,17)(2,10)(4,14,22,31)(5,16,6,15)(8,11)(9,25)(12,24)(13,28)(18,23)(19,20,30,21)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=efg,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251684, #72251691, #72251697, … |
| maximal normal subgroup of | #72251684, #72251691, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353439);