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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 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)(15,23,24,27)(16,25,28,20)(17,21,18,26);(7,8)(10,11)(13,15,17,25,19,24,18,20)(14,16,26,27,22,28,21,23);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,28)(14,20,22,25)(15,21,24,26)(17,27,18,23);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,21,22,26)(15,20,24,25)(16,27,28,23);(6,12)(10,11)(14,22)(16,28)(21,26)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6395027);