Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,30)(21,31)(22,27)(23,32)(24,28)(26,29);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,24,29)(18,25,28,26)(20,22,21,23)(27,31,32,30);(9,12,13,10)(11,15,16,14)(17,20,23,19,24,21,22,29)(18,25,27,30,28,26,32,31);(11,16)(14,15)(17,21,23,29,24,20,22,19)(18,26,27,31,28,25,32,30);(1,3,4,8)(2,6,5,7)(17,22,24,23)(18,27,28,32)(19,21,29,20)(25,31,26,30)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=h,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511324);