Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,19,28)(8,21,24,12)(9,13,25,14)(11,20,27,18)(17,33,34,32)(22,31,23,30)(26,36,29,35);(2,4)(5,7)(6,11)(8,17)(9,20)(10,19)(12,29)(13,16)(14,28)(15,27)(18,25)(21,26)(22,35)(23,36)(24,34)(30,33)(31,32);(1,2)(3,4)(5,8,10,24)(6,12,15,21)(7,17,19,34)(9,22,25,23)(11,26,27,29)(13,30,14,31)(16,32,28,33)(18,36,20,35);(2,4)(6,13)(7,18)(8,22)(9,25)(11,27)(14,15)(17,34)(19,20)(23,24)(26,33)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383503);