Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1001 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)(5,6,9,13,10,14,23,29)(7,11,16,25,17,26,33,36)(8,18,21,27,20,28,22,12)(15,30,31,35,19,34,32,24);(5,7)(6,11)(8,19,20,15)(9,16)(10,17)(12,24,27,35)(13,25)(14,26)(18,30,28,34)(21,32,22,31)(23,33)(29,36);(5,8,9,22,10,20,23,21)(6,12,13,28,14,27,29,18)(7,15,16,32,17,19,33,31)(11,24,25,34,26,35,36,30);(2,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,24)(13,25)(14,26)(18,30)(19,20)(21,31)(22,32)(23,33)(27,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332342);