Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(7,12)(14,17)(15,19)(23,27)(25,28);(5,6,9,10)(7,11,12,8)(14,17)(18,20)(21,22,24,26)(23,25,27,28);(13,14,16,17)(15,18,19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,25,26,28);(5,8,9,11)(6,12,10,7)Polycyclic presentation:
2^9|b^2=fgh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 685824);