Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,15,20,16,21,13,19)(11,22,26,34,27,40,24,39)(12,28,14,31,30,18,29,38)(17,33,36,23,37,32,35,25);(9,11,16,27)(10,17,21,37)(12,23,30,25)(13,24,15,26)(14,32,29,33)(18,34,28,39)(19,35,20,36)(22,38,40,31);(1,3,4,8,5,2,6,7)(9,12,15,14,16,30,13,29)(10,18,20,38,21,28,19,31)(11,23,26,32,27,25,24,33)(17,34,36,40,37,39,35,22);(2,3)(7,8)(9,13,16,15)(10,19,21,20)(11,24,27,26)(12,29,30,14)(17,35,37,36)(18,31,28,38)(22,39,40,34)(23,33,25,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234338, #72234343, #72234346, … |
| maximal normal subgroup of | #72234338, #72234343, #72234346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391288);