Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34925 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,17,13,15,22,23)(11,18,20,21,19,16,24,12);(1,2)(3,4)(10,15)(11,19)(17,23)(20,24);(6,8)(9,11)(10,16)(12,17)(13,19)(14,20)(15,18)(21,23)(22,24);(2,4)(10,15)(11,19)(17,23)(20,24);(9,12,14,18,13,21,22,16)(10,11,17,20,15,19,23,24)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=dfi,d^c=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72343927, #72343931, #72343972 |
| maximal normal subgroup of | #72343927, #72343931, #72343972 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290972);