Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33669 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(2,7)(3,12)(4,15)(5,18)(8,21)(9,20)(10,25)(11,22)(13,26)(14,17)(16,31)(19,30)(23,29)(24,32)(27,28);(3,13)(7,20)(12,26)(14,29)(16,27)(21,22)(25,32)(28,31);(1,2)(3,14)(4,17)(5,8)(6,9)(7,12)(10,15)(11,18)(13,29)(16,32)(19,23)(20,26)(21,31)(22,28)(24,30)(25,27);(2,8,9,11)(3,13)(7,21,20,22)(10,23,24,17)(12,26)(14,25,29,32)(16,27)(28,31)Polycyclic presentation:
2^9|d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,f^b=fi,d^c=dh,e^c=ei,f^c=fg,g^c=gh,h^c=hi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214100, #72214568, #72214585, … |
| maximal normal subgroup of | #72214100, #72214568, #72214585, … |
| maximal quotient of | #72216258, #72224983, C2×#513022 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420218);