Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,9,16,22)(8,12,20,28)(10,14)(11,13)(15,33)(17,23,25,30)(18,36)(19,34)(21,29,26,31)(24,32)(27,35);(5,7)(6,9)(8,18)(10,16)(11,25)(12,27)(13,30)(14,22)(15,21)(17,24)(19,29)(20,35)(23,32)(26,34)(28,36)(31,33);(2,4)(5,8,10,20)(6,12,14,28)(7,15,16,34)(9,19,22,33)(11,26,24,21)(13,31,32,29)(17,35,25,18)(23,36,30,27);(5,9)(6,7)(8,19)(10,22)(11,23)(12,15)(13,17)(14,16)(18,29)(20,33)(21,27)(24,30)(25,32)(26,36)(28,34)(31,35)Polycyclic presentation:
2^9|a^2=ef,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302625);