Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| 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 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15)(7,17,19,12)(8,22)(10,16,11,14)(13,25)(18,27)(20,29,21,28)(23,36)(24,32)(26,31)(30,35)(33,34);(1,2)(3,4)(5,7)(6,12)(8,23,25,35)(9,19)(10,20)(11,21)(13,30,22,36)(14,29)(15,17)(16,28)(18,24,34,26)(27,32,33,31);(2,4)(5,8)(6,13,15,22)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(14,31,16,32)(19,34)(20,35)(21,23)(28,30,29,36);(6,14)(7,19)(10,11)(12,28)(13,31)(15,16)(17,29)(18,34)(22,32)(24,26)(27,30)(33,36)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255644, #72256181, #72256186, … |
| maximal normal subgroup of | #72255644, #72256181, #72256186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330623);