Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,26,21,20)(19,27,31,28,29,25,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,26)(21,23)(25,32)(27,28)(30,31);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,25,26,27)(20,28,24,32)(21,30,23,31);(11,16)(13,15)(17,20,23,18,22,24,21,26)(19,28,31,25,29,32,30,27);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(18,26)(19,29)(20,24)(30,31)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=hi,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6995290);