Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
904 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 904 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24)(25,26,29,33,30,34,28,37)(27,31,35,40,36,32,38,39);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,36)(26,31,34,32)(28,38,29,35)(33,40,37,39);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,27)(26,32)(28,35)(29,38)(30,36)(31,34)(33,40)(37,39);(10,15)(14,20)(16,22)(21,24)(25,28,30,29)(26,33,34,37)(27,35,36,38)(31,39,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96929);