Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #492512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33840 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,16)(13,26)(14,17)(18,23)(19,24)(20,28)(21,31)(22,38)(25,32)(27,34)(29,37)(30,35)(33,39)(36,40);(1,2,5,7)(3,6,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,31)(17,28)(18,29)(19,30)(23,37)(24,35)(25,34)(26,38)(27,32)(33,40)(36,39);(10,16)(11,20)(12,23)(13,24)(15,27)(17,32)(18,19)(21,36)(25,39)(26,33)(28,31)(29,38)(30,40)(34,37);(1,3)(4,5)(6,8)(10,17)(15,28)(16,32)(18,33)(19,26)(27,31)(29,40)(30,38)Polycyclic presentation:
2^9|b^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg65 conjugacy classes — view the full character table.
| maximal subgroup of | #72309684 |
| maximal normal subgroup of | #72309684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399798);