Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153179 |
| 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 |
| prime radical | #3018 | C22 |
| 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 | |
542 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 542 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:
(1,2,8,3)(4,15,20,27)(5,12,7,10)(6,22,24,28)(9,19,13,17)(11,25,14,21)(16,18,26,23)(29,30,32,31);(1,3,8,2)(4,16,20,26)(5,10,7,12)(6,11,24,14)(9,18,13,23)(15,29,27,32)(17,30,19,31)(21,28,25,22);(1,4)(2,9)(3,13)(5,19)(6,23)(7,17)(8,20)(10,27)(11,16)(12,15)(14,26)(18,24)(21,32)(22,30)(25,29)(28,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60378);