Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| 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 | #37190 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1135 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(2,3)(5,6)(7,15)(8,20)(9,24)(10,12)(11,17)(13,23)(14,27)(16,33)(18,34)(19,29)(21,28)(22,30)(25,35)(26,32)(31,36);(1,2)(3,4)(5,6)(7,16)(8,13)(9,24)(10,12)(11,18)(14,27)(15,33)(17,34)(19,31)(20,23)(21,35)(22,32)(25,28)(26,30)(29,36);(1,3,4,2)(5,7)(6,11)(8,21)(9,25)(10,19)(12,28)(13,29)(14,31)(15,20)(16,24)(17,23)(18,27)(22,33)(26,36)(30,34)(32,35);(6,12)(11,28)(13,20)(14,24)(15,29)(16,31)(30,32)(34,35)Polycyclic presentation:
2^9|c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406308);