Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| 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×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 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)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,25)(19,27,20,29)(21,31,32,26)(23,30,24,28);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,22)(21,30)(24,25)(26,29)(27,31)(28,32);(10,14)(15,16)(17,20,22,19)(18,24,25,23)(21,28,32,30)(26,27,31,29);(9,11,13,12)(10,15,14,16)(19,20)(23,24)(27,29)(28,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,25)(19,20)(26,31)(28,30)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392842);