Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2396 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)(5,6,10,9)(7,12,8,11)(21,22,26,24)(23,25,28,27);(5,6,10,9)(7,12,8,11)(13,14,17,16)(15,19,20,18)(21,23,26,28)(22,27,24,25);(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,18,16,19)(21,24,26,22)(23,25,28,27);(1,2)(3,4)(5,8,10,7)(6,12,9,11);(5,9,10,6)(7,12,8,11)(13,16,17,14)(15,19,20,18)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757727);