Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238303 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18)(19,22)(20,28)(21,25)(23,27)(24,30)(26,31)(29,32);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19)(18,22)(20,27)(21,26)(23,28)(24,32)(25,31)(29,30);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,19)(18,23)(20,29)(21,26)(22,28)(24,31)(25,32)(27,30);(4,13)(7,11)(9,12)(14,16)(19,26)(22,31)(23,32)(27,29)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,g^b=gh,d^c=di,e^c=eh,f^c=fhi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145499);