Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| 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×D4 | C24 |
| prime radical | #23913 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1205 conjugacy classes of subgroups — too many to draw. Only its 140 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)(8,12)(9,14,24,13)(10,15)(11,19,27,22)(17,31)(18,26)(20,29,23,28)(21,30)(25,33,36,34)(32,35);(1,2)(3,4)(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,16)(14,26)(15,27)(19,24)(20,34)(21,32)(23,33)(28,31)(29,35)(30,36);(2,4)(5,8,10,21)(6,12,15,30)(7,17,18,32)(9,23,24,20)(11,25,27,36)(13,28,14,29)(16,31,26,35)(19,34,22,33);(6,13)(7,18)(9,24)(11,26)(12,28)(14,15)(16,27)(17,32)(20,23)(25,35)(29,30)(31,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72284358, #72284368, #72285468, … |
| maximal normal subgroup of | #72284358, #72284368, #72285468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408294);