Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | #37271 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,16)(15,21,22,28)(18,26,23,20)(19,27,25,24);(2,4)(7,8)(10,12)(14,16)(15,22)(20,26)(24,27);(5,7,9,8)(6,10,11,12)(13,15,23,27,17,22,18,24)(14,19,26,28,16,25,20,21);(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,19,22,25)(18,26,23,20)(21,27,28,24);(6,11)(10,12)(14,16)(19,25)(20,26)(21,28)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390361);