Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,15)(8,9)(13,14)(16,19)(17,18);(1,2)(3,4)(5,6)(7,16)(8,13)(9,11)(10,14)(12,17)(15,20)(18,19);(2,4)(5,7)(6,12,14,20)(8,15)(9,17)(10,18)(11,19,13,16);(6,13)(8,9)(11,14)(12,16)(15,17)(19,20)Polycyclic presentation:
2^9|c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250755, #72250784, #72250789, … |
| maximal normal subgroup of | #72250755, #72250784, #72250789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330382);