Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #47031 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1078 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,11)(4,15)(8,23)(9,13)(10,22)(12,20)(14,29)(16,17)(19,24)(21,32)(25,30)(26,28)(27,31);(2,8,9,24)(3,12)(4,16)(7,19,13,23)(10,25,26,32)(11,20)(14,31)(15,17)(21,28,30,22)(27,29);(1,2,6,10)(3,13,12,28)(4,8,17,25)(5,9,18,26)(7,20,22,11)(14,30,29,23)(15,32,16,24)(19,27,21,31);(3,11)(7,13)(12,20)(14,27)(19,23)(21,30)(22,28)(29,31)Polycyclic presentation:
2^9|b^2=e,c^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237836, #72237838, #72237843, … |
| maximal normal subgroup of | #72237836, #72237838, #72237843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159133);