Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14)(7,15,18,29)(8,19,22,12)(9,23,24,13)(11,20,28,17)(16,33,25,36)(21,30,26,32)(27,34,31,35);(1,2,4,3)(7,16)(9,24)(11,27)(13,23)(15,33)(17,35)(18,25)(20,34)(21,26)(28,31)(29,36)(30,32);(5,7)(6,11)(8,20)(9,25)(10,18)(12,29)(13,31)(14,28)(15,19)(16,24)(17,22)(21,35)(23,27)(26,34)(30,33)(32,36);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407980);