Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15610 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
951 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)(3,12)(4,8,16,26)(5,11)(6,10)(7,14)(9,17,27,19)(13,24,30,21)(15,28)(18,23)(20,25)(22,32,29,31);(1,3)(2,7,10,12)(4,13,16,30)(5,18)(6,14)(8,24)(9,22)(11,28,25,23)(15,20)(17,32,19,31)(21,26)(27,29);(1,4)(2,8,10,26)(3,13,14,30)(5,19)(6,16)(7,21)(9,25,27,11)(12,24)(15,32,18,31)(17,20)(22,23)(28,29);(1,4)(2,9,10,27)(3,13,14,30)(5,17)(6,16)(7,22)(8,25,26,11)(12,29)(15,31,18,32)(19,20)(21,23)(24,28)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344908);