Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(6,8)(9,10,14,18,15,19,13,28)(11,16,21,32,22,33,29,31)(12,23,27,35,25,37,26,17)(20,34,24,30,36,40,38,39);(1,2,4,3)(9,11)(10,16)(12,24)(13,29)(14,21)(15,22)(17,34)(18,32)(19,33)(20,26)(23,30)(25,38)(27,36)(28,31)(35,40)(37,39);(5,6,7,8)(9,12)(10,17,19,35)(11,20,22,36)(13,26)(14,27)(15,25)(16,30)(18,23,28,37)(21,24,29,38)(31,34)(32,40)(33,39);(1,3,4,2)(12,25)(17,35)(20,36)(23,37)(24,38)(26,27)(30,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281576);