Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,19,20,17)(16,18);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,17)(16,18)(19,20);(2,4)(13,14)(15,17)(16,18)(19,20);(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=e,b^2=ef,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72260933, #72261082, #72261094, … |
| maximal normal subgroup of | #72260933, #72261082, #72261094, … |
| maximal quotient of | #72261416 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303729);