Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196110 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11,15,19)(8,10)(9,13)(12,16)(14,17,20,18);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(14,20)(18,19);(2,4)(5,7)(6,11,13,19)(8,14)(9,15)(10,17,12,18)(16,20);(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72216203, #72216217, … |
| maximal normal subgroup of | #72214076, #72216203, #72216217, … |
| maximal quotient of | C2×#374114, #72236747, #72239583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281310);