Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(14,19)(15,21)(17,25)(24,27);(13,14)(15,20)(16,21)(17,22)(18,19)(23,27)(24,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,23)(18,21)(19,20)(22,24)(25,28)(26,27);(7,11)(10,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,26)(25,28);(5,7,8,11)(6,10,9,12)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23);(5,6)(7,10)(8,9)(11,12)(13,17,18,25)(14,22,19,26)(15,23,21,28)(16,24,20,27)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533795);