Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1629 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
909 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 909 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12,15,30)(4,6)(5,17)(8,22,24,31)(9,21)(10,18)(13,19,14,29)(16,20)(23,28,25,26)(27,32);(1,2)(3,8)(4,9)(5,18)(6,21)(7,11)(10,17)(12,28)(13,23)(14,25)(15,24)(16,27)(19,31)(20,32)(22,29)(26,30);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,23)(10,26)(11,24)(12,16)(17,30)(18,31)(20,29)(21,25)(22,32)(27,28);(1,4,7,6)(2,9,11,21)(3,13,15,14)(5,16,20,17)(8,23,24,25)(10,18,27,32)(12,29,30,19)(22,26,31,28)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72214352, #72214424, … |
| maximal normal subgroup of | #72214208, #72214352, #72214424, … |
| maximal quotient of | #72224804, #72226781, #72226784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396125);