Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480268 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3322 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)(8,13)(9,21,22,11)(10,14)(15,25)(16,30,31,23)(17,26)(18,27)(19,33,34,24)(20,28)(29,36,32,35);(6,11)(12,23)(13,24)(14,21)(25,35)(26,30)(27,36)(28,33);(2,4)(5,7,10,17)(6,12,14,26)(8,18,20,15)(9,16,22,31)(11,23,21,30)(13,27,28,25)(19,32,34,29)(24,36,33,35);(5,8,10,20)(6,13,14,28)(7,15,17,18)(9,19,22,34)(11,24,21,33)(12,25,26,27)(16,29,31,32)(23,35,30,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303197, #72303216, #72303429, … |
| maximal normal subgroup of | #72303197, #72303216, #72303429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400622);