Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401009 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,20)(9,14)(10,24)(11,15)(13,23)(16,25)(17,31)(18,27)(19,28)(21,35)(22,29)(26,30)(32,34)(33,36);(2,4)(5,7,11,19)(6,12,15,28)(8,17,21,32)(9,16,10,18)(13,26,29,36)(14,25,24,27)(20,34,35,31)(22,30,23,33);(7,16)(9,10)(12,25)(14,24)(17,30)(18,19)(20,35)(22,23)(26,34)(27,28)(31,36)(32,33);(5,8)(6,13,15,29)(7,17)(9,22)(10,23)(11,21)(12,26,28,36)(14,20,24,35)(16,30)(18,33)(19,32)(25,34,27,31)Polycyclic presentation:
2^9|b^2=i,d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256515, #72256519, #72256521 |
| maximal normal subgroup of | #72256515, #72256519, #72256521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321098);