Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45332 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(10,13)(14,16)(17,18)(19,29)(20,26)(21,32)(22,28)(23,25)(24,31)(27,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,26)(22,30)(23,31)(25,29)(28,32);(2,3)(6,7)(18,25)(20,28)(24,29)(27,32);(17,20,23,28)(18,26,25,22)(19,27,31,32)(21,29,30,24);(1,3,5,2)(4,7,8,6)(17,21,23,30)(18,27,25,32)(19,26,31,22)(20,29,28,24);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,22,23,26)(18,28,25,20)(19,30,31,21)(24,32,29,27)Polycyclic presentation:
2^9|d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625443);