Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16152 |
| 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 | |
651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,14,6,13)(3,15,19,30,20,29,18,9)(4,21,22,17,24,31,23,10)(5,25,28,16,27,11,26,32);(1,3,7,20)(2,9)(4,11,24,25)(5,21)(6,18,8,19)(10,26)(12,15)(13,29)(14,30)(16,22,32,23)(17,28)(27,31);(1,4,6,22,7,24,8,23)(2,10,12,31,14,17,13,21)(3,16,18,11,20,32,19,25)(5,15,26,30,27,29,28,9);(1,5,7,27)(2,11)(3,17)(4,15,24,29)(6,26,8,28)(9,22,30,23)(10,20)(12,32)(13,16)(14,25)(18,21)(19,31)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,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,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361044);