Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(22,25)(24,32)(28,31)(29,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,28)(29,31)(30,32);(11,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(11,15)(13,16)(17,20,21,26)(18,25,23,22)(19,28,29,32)(24,27,31,30);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,23)(20,26)(24,31)(28,32)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6379635);