Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15310 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1887 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,15)(9,20)(14,29)(16,24)(17,28)(19,32)(21,27)(22,26)(23,31)(25,30);(1,3,5,15,7,17,6,16)(2,8,10,13,12,24,11,28)(4,19,20,31,22,30,21,14)(9,25,18,23,27,32,26,29);(1,4,6,20,7,22,5,21)(2,9,11,18,12,27,10,26)(3,14,15,30,17,31,16,19)(8,23,13,25,24,29,28,32);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,23)(10,28)(11,13)(12,24)(18,25)(19,21)(20,30)(22,31)(26,32)(27,29)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72316368, C2×#422909, #72339020, … |
| maximal normal subgroup of | #72316368, C2×#422909, #72339020, … |
| maximal quotient of | C2×#422909, #72316826, #72316836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330105);