Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394482 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2435 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14820 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,5,9,22,11)(3,12,16,24,14,7,15,25)(4,17,19,27,21,28,20,8)(13,26,18,29,31,32,30,23);(1,3,5,14)(2,7,9,12)(4,18)(6,16,22,15)(8,26)(10,25,11,24)(13,20)(17,29)(19,31)(21,30)(23,28)(27,32);(1,4)(2,8)(3,13)(5,21)(6,20)(7,23)(9,27)(10,28)(11,17)(12,29)(14,31)(15,18)(16,30)(19,22)(24,26)(25,32);(1,2)(3,7)(4,8)(5,9)(6,10)(11,22)(12,14)(13,23)(15,24)(16,25)(17,19)(18,26)(20,28)(21,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319859, #72319954, #72319976, … |
| maximal normal subgroup of | #72319859, #72319954, #72319976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308826);