Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2716 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)(7,12)(8,10)(9,11)(13,14,18,21,19,24,16,23)(15,25,26,17,22,27,28,20);(1,2)(3,4)(13,15,19,22)(14,20,24,17)(16,26,18,28)(21,27,23,25);(2,4)(5,7,9,8)(6,10,11,12)(13,16,19,18)(14,21,24,23)(15,26,22,28)(17,25,20,27);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,20)(14,22,24,15)(16,25,18,27)(21,28,23,26);(13,16,19,18)(14,23,24,21)(15,26,22,28)(17,27,20,25)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359236);