Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4×#95 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,21,23,16)(12,17)(13,19)(14,20)(15,18)(22,33,26,30)(24,32,25,31)(27,35)(28,36)(29,34)(37,40,38,39);(1,2,4,8,5,7,3,6)(9,11,13,24,15,23,14,25)(10,16,19,31,18,21,20,32)(12,26,27,38,29,22,28,37)(17,33,35,40,34,30,36,39);(2,7)(6,8)(9,12,14,28,15,29,13,27)(10,17,20,36,18,34,19,35)(11,22,25,38,23,26,24,37)(16,30,32,40,21,33,31,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,g^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224831, #72224912, #72224914, … |
| maximal normal subgroup of | #72224831, #72224912, #72224914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12518);