Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,21,28)(8,22,26,31)(9,15,10,14)(12,19,29,20)(13,24,32,25)(18,27,34,36)(23,35,33,30);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,27)(24,34)(26,33)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18,21,34)(9,25)(10,24)(11,26)(12,27,29,36)(14,22)(15,31)(16,32)(17,30,28,35)(19,33,20,23);(7,19)(9,10)(12,28)(14,15)(17,29)(18,33)(20,21)(22,31)(23,34)(24,25)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=i,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250755, #72250788, … |
| maximal normal subgroup of | #72250742, #72250755, #72250788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330384);