Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,8,4)(10,13)(11,15)(17,18,21,23)(19,24,28,30)(20,26,22,25)(27,32,29,31);(1,3,6,8)(2,4,7,5)(9,10,12,13)(11,14,15,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,31,26,32);(1,5,6,4)(2,3,7,8)(10,13)(14,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32);(17,20,21,22)(18,26,23,25)(19,27,28,29)(24,32,30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417792);