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 | #37232 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(14,16)(17,18,22,23)(19,24,25,20)(21,26,30,31)(27,32,29,28);(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,29)(22,25)(23,24)(26,32)(27,30)(28,31);(1,2,4,6)(3,5,8,7)(9,11,12,15)(10,14,13,16)(18,23)(19,25)(21,30)(28,32);(1,2)(3,5)(4,6)(7,8)(17,19)(18,24)(20,23)(21,27)(22,25)(26,32)(28,31)(29,30);(1,3)(2,5)(4,8)(6,7)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27);(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,32)(25,29)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7738010);