Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,21,25,22)(20,24,30,32)(27,28,31,29);(1,2,3,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,26,23,25,22,18)(20,27,28,32,30,31,29,24);(6,8)(10,14)(13,16)(17,20,23,30)(18,24,26,32)(19,27,25,31)(21,28,22,29);(2,4)(18,19)(21,22)(24,27)(25,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328216);