Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1630 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,24,26,35)(13,19)(14,29)(15,18)(17,27,34,28)(20,33,36,40)(21,32)(22,31)(23,38)(25,30,37,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,25,26,37)(13,22)(14,23)(15,21)(17,33,34,40)(18,32)(19,31)(20,28,36,27)(24,39,35,30)(29,38);(5,8)(9,12)(10,17)(11,20)(13,28)(14,27)(15,26)(16,30)(18,34)(19,35)(21,36)(22,25)(23,37)(24,29)(31,40)(32,39)(33,38);(10,18)(11,21)(17,34)(19,29)(20,36)(22,23)(24,35)(25,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255848, #72255859, #72255873, … |
| maximal normal subgroup of | #72255848, #72255859, #72255873, … |
| maximal quotient of | #72262593, #72262784, #72266273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145254);