Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469776 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1371 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)(7,11)(8,10)(9,13)(12,16)(14,17)(15,19)(18,20);(6,10,13,12)(11,17,19,18);(7,14)(8,16)(10,12)(11,18)(15,20)(17,19);(2,4)(5,7,8,14,9,15,16,20)(6,11,10,17,13,19,12,18)Polycyclic presentation:
2^9|b^2=g,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250767, #72250784, #72250789, … |
| maximal normal subgroup of | #72250767, #72250784, #72250789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377477);