Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1898 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,20)(9,14)(10,25)(11,17)(12,19)(13,18)(15,24)(21,27)(22,33)(23,28)(26,31)(29,36)(30,35)(32,34);(1,2)(3,4)(5,7,11,10)(6,12,17,28)(8,18,22,26)(9,23,24,19)(13,29,31,27)(14,16,15,25)(20,32,33,30)(21,35,36,34);(2,4)(5,8,11,22)(6,13,17,31)(7,18,10,26)(9,21,24,36)(12,27,28,29)(14,30,15,32)(16,33,25,20)(19,34,23,35);(6,14)(7,10)(9,24)(12,25)(13,30)(15,17)(16,28)(18,26)(20,27)(21,36)(29,33)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407066);