Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(10,13)(14,16)(19,23)(21,24)(25,26)(27,30)(28,29)(31,32);(2,4)(10,13)(11,15)(19,23)(21,24)(26,29)(27,31);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,32)(28,29)(30,31);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=efg,c^2=fh,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72328618, #72328751, #72328904, … |
| maximal normal subgroup of | #72328618, #72328751, #72328904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67368);