Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153155 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | #532 | C23 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 32 points:
(1,2)(3,13)(4,8)(5,19)(6,10)(7,12)(9,15)(11,18)(14,25)(16,27)(17,26)(20,23)(21,28)(22,29)(24,31)(30,32);(2,8)(3,14)(4,6)(5,20)(9,24)(10,12)(11,19)(13,25)(15,31)(16,30)(17,21)(23,29)(26,28)(27,32);(1,3)(2,9)(4,16)(5,21)(6,15)(7,17)(8,13)(10,26)(11,28)(12,27)(14,20)(18,30)(19,24)(22,31)(23,32)(25,29)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72283193, #72283987 |
| maximal normal subgroup of | #72283193, #72283987 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60351);