Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16584 |
| 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 | |
681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,18,29,19,28,17,9)(4,20,22,27,16,31,26,10)(5,23,21,30,24,11,25,32);(1,3,6,17,8,19,7,18)(2,9,12,15,14,29,13,28)(4,21,26,5,16,25,22,24)(10,30,20,11,27,32,31,23);(1,4,6,22,8,16,7,26)(2,10,12,31,14,27,13,20)(3,5,17,21,19,24,18,25)(9,23,15,32,29,11,28,30);(1,5,8,24)(2,11)(3,16,19,4)(6,25,7,21)(9,27)(10,29)(12,32)(13,30)(14,23)(15,31)(17,22,18,26)(20,28)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374888);