Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#546 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18)(9,22,23,13)(10,14)(12,20,28,36)(15,29)(16,31,32,25)(17,26)(19,33)(21,34)(24,30,27,35);(1,2)(3,4)(5,7,10,17)(6,11,14,26)(8,19,21,15)(9,16,23,32)(12,27,28,24)(13,25,22,31)(18,33,34,29)(20,35,36,30);(5,8,10,21)(6,12,14,28)(7,15,17,19)(9,20,23,36)(11,24,26,27)(13,18,22,34)(16,30,32,35)(25,29,31,33);(2,4)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72312239, #72312247, #72312266, … |
| maximal normal subgroup of | #72312239, #72312247, #72312266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404555);