Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#381 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4437 conjugacy classes of subgroups; only the 1296 normal subgroups are shown.
This group has 4437 conjugacy classes of subgroups — too many to draw. Only its 1296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11,14,7)(4,15)(5,9)(6,10)(8,17)(12,19,26,23)(13,24,27,20)(16,28)(18,22)(21,29)(25,30,32,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,23)(9,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,22)(11,24)(12,25)(14,27)(15,28)(17,29)(19,30)(23,31)(26,32);(5,18)(9,22)(13,27)(16,29)(20,24)(21,28)(25,32)(30,31)Polycyclic presentation:
2^9|a^2=h,c^2=f,b^a=bf,c^a=cg,f^a=fi,h^b=hi,g^c=gi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72266788, #72266791, #72266795, … |
| maximal normal subgroup of | #72266788, #72266791, #72266795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278093);