Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35292 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,22)(15,17,21,18)(16,24,27,20)(23,28,25,26);(7,8)(10,11)(13,15,18,14,19,21,17,22)(16,25,28,24,27,23,26,20);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,27)(14,20,22,24)(15,23,21,25)(17,26,18,28);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,21,22,15)(16,26,27,28)(20,25,24,23);(6,12)(10,11)(14,22)(15,21)(16,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394979);