Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14,17,20)(15,19,18,16);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,18)(19,20)(22,25)(26,28);(13,16)(14,18)(15,20)(17,19)(21,22,24,25)(23,26,27,28);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,19)(18,20);(5,6)(7,9)(8,11)(10,12)(13,15)(14,19)(16,20)(17,18)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,e^2=gh,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424861);