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 | #43155 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(10,13)(14,15)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2,5,6)(3,7,4,8)(18,23)(20,27)(25,29)(28,32);(1,3,5,4)(2,7,6,8)(9,10,12,13)(11,15,16,14)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(9,11,12,16)(10,14,13,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,32)(30,31);(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589736);