Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C27 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
9536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,17)(16,18)(19,20);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,19)(18,20);(17,20);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72217576, #72217617, #72222882, … |
| maximal normal subgroup of | #72217576, #72217617, #72222882, … |
| maximal quotient of | #72222816, #72227675, #72255632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301608);