Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
951 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)(7,19)(8,23,28,33)(9,29)(10,17)(11,16)(12,18)(13,21)(14,26,34,27)(15,22)(20,30,35,36)(24,32,25,31);(5,7,12,22)(6,13,18,29)(8,24,28,25)(9,11,21,10)(14,31,34,32)(15,17,19,16)(20,27,35,26)(23,36,33,30);(5,8,12,28)(6,14)(7,20,22,35)(9,25,21,24)(10,26,11,27)(13,30)(15,32)(16,33)(17,23)(18,34)(19,31)(29,36);(2,4)(5,9,10,7,12,21,11,22)(6,15,16,13,18,19,17,29)(8,25,26,20,28,24,27,35)(14,32,33,30,34,31,23,36)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,g^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=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341666);