Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14336 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups; only the 352 normal subgroups are shown.
This group has 2281 conjugacy classes of subgroups — too many to draw. Only its 352 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 — 4 generators on 32 points:
(1,2)(3,7)(4,14,17,8)(5,9)(6,10)(11,26,15,20)(12,21)(13,22)(16,24,18,23)(19,25)(27,31,28,30)(29,32);(1,3)(2,7)(4,15,17,11)(5,12)(6,13)(8,20,14,26)(9,21)(10,22)(16,28,18,27)(19,29)(23,30,24,31)(25,32);(1,4)(2,8)(3,11)(5,18,19,16)(6,17)(7,20)(9,24,25,23)(10,14)(12,28,29,27)(13,15)(21,31,32,30)(22,26);(1,5)(2,9)(3,12)(4,16)(6,19)(7,21)(8,23)(10,25)(11,27)(13,29)(14,24)(15,28)(17,18)(20,30)(22,32)(26,31)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72266907, #72266960 |
| maximal normal subgroup of | #72266907, #72266960 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289668);