Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #394 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,14,20,15,19,13,18)(11,21,25,16,24,32,23,31)(12,17,29,34,28,35,27,33)(22,37,39,40,26,36,38,30);(2,4)(5,6,7,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,30)(18,32)(19,31)(20,21)(22,28)(27,38)(29,39)(33,37)(34,36)(35,40);(9,12)(10,17)(11,22,24,26)(13,27)(14,29)(15,28)(16,30,31,40)(18,33)(19,35)(20,34)(21,36,32,37)(23,38,25,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314514, #72314520, #72314528, … |
| maximal normal subgroup of | #72314514, #72314520, #72314528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31232);