Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418492 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,21)(9,24)(10,23)(11,14)(12,18)(13,26)(16,20)(17,29)(19,30)(22,28)(25,33)(27,32)(31,35)(34,36);(1,2,3,4)(5,7,10,18,11,20,21,19)(6,12,8,15,14,30,23,16)(9,25,26,34,28,35,27,17)(13,31,24,36,32,33,22,29);(5,8,21,6,11,23,10,14)(7,16,18,12,20,15,19,30)(9,22,26,13,28,24,27,32)(17,31,35,29,34,33,25,36);(2,4)(5,9,11,28)(6,13,14,32)(7,17,20,34)(8,22,23,24)(10,27,21,26)(12,29,30,36)(15,33,16,31)(18,35,19,25)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330104);