Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(18,22)(20,28)(23,29)(24,30)(25,31)(27,32);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,18,20,22,21,25,28,31)(19,23,27,29,26,24,32,30);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,25)(28,32)(30,31);(11,15)(13,16)(17,19)(18,24)(20,27)(21,26)(22,30)(23,25)(28,32)(29,31);(19,26)(23,24)(27,32)(29,30)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368585);