Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,28)(25,27);(5,6,8,9)(7,10,11,12)(13,15,18,16)(14,19,20,21)(17,23,25,24)(22,26,27,28);(5,7,8,11)(6,10,9,12)(14,20)(19,21)(22,27)(26,28);(13,16,18,15)(14,21,20,19)(17,23,25,24)(22,26,27,28);(13,17,18,25)(14,22,20,27)(15,23,16,24)(19,26,21,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532988);