Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11,16,20)(8,14)(9,12)(10,13)(15,19,17,18);(1,2)(3,4)(7,15,16,17)(11,18,20,19);(2,4)(5,7)(6,11,13,20)(8,17)(9,15)(10,16)(12,19,14,18);(6,12,13,14)(11,19,20,18)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72338514, #72338521, #72338522, … |
| maximal normal subgroup of | #72338514, #72338521, #72338522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308674);