Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,15)(9,12)(11,17)(13,19)(14,18)(16,20);(10,18)(19,20);(1,2,3,4)(5,7)(6,10)(8,16,17,13)(9,14)(11,20,15,19)(12,18);(5,8)(6,11,12,15)(7,13)(9,17)(10,19,18,20)(14,16)Polycyclic presentation:
2^9|c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225669, #72225681, #72225682, … |
| maximal normal subgroup of | #72225669, #72225681, #72225682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376874);