Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,11,13)(7,9,15,10)(8,16,19,14)(12,17,20,18);(5,7,9,13,11,15,10,6)(8,12,17,14,19,20,18,16);(5,8)(6,12)(7,14)(9,17)(10,18)(11,19)(13,20)(15,16);(1,2,3,4)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215866, #72216141, #72219485, … |
| maximal normal subgroup of | #72215866, #72216141, #72219485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318994);