Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425091 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,13,10,14,23,21)(7,11,16,25,17,26,22,28)(8,12,19,18,20,27,35,33)(15,24,30,29,31,36,34,32);(1,2)(3,4)(5,6)(7,11)(8,18)(9,21)(10,14)(12,19)(13,23)(15,29)(16,28)(17,26)(20,33)(22,25)(24,30)(27,35)(31,32)(34,36);(2,4)(5,7,10,17)(6,11,14,26)(8,15,20,31)(9,22,23,16)(12,24,27,36)(13,28,21,25)(18,32,33,29)(19,34,35,30);(5,8,10,20)(6,12)(7,15,17,31)(9,19,23,35)(11,24)(13,18)(14,27)(16,30,22,34)(21,33)(25,29)(26,36)(28,32)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334621);