Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16786 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,17,8)(4,9,5,20)(6,12)(7,11)(10,21,26,19)(14,25)(15,27)(16,22)(18,29,30,24)(23,32)(28,31);(1,3,14,30,6,17,7,18)(2,8,11,24,12,13,25,29)(4,15,21,32,5,16,19,31)(9,22,10,23,20,27,26,28);(1,4)(2,9)(3,15,17,16)(5,6)(7,21)(8,22,13,27)(10,25)(11,26)(12,20)(14,19)(18,32,30,31)(23,24,28,29);(1,5,6,4)(2,10)(3,16,17,15)(7,21,14,19)(8,23)(9,25)(11,20)(12,26)(13,28)(18,32,30,31)(22,29)(24,27)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=ef,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378480);