Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14919 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1101 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,14)(12,23)(13,20,27,24)(15,21,28,29)(16,22)(19,26)(25,31,32,30);(1,3)(2,7,10,11)(4,12,16,26)(5,13)(6,14)(8,19)(9,20,17,24)(15,25,28,32)(18,27)(21,30)(22,23)(29,31);(1,4)(2,8)(3,12,14,26)(5,15)(6,16)(7,19,11,23)(9,21)(10,22)(13,25,27,32)(17,29)(18,28)(20,30,24,31);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,21)(10,17)(11,24)(12,25)(14,27)(16,28)(19,30)(22,29)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237789, #72237793, #72237799, … |
| maximal normal subgroup of | #72237789, #72237793, #72237799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313047);