Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,19,22,15)(12,16,24,31)(13,25)(14,18)(17,26)(20,27)(21,32,35,28)(23,36,37,30)(29,34)(33,38)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,32)(18,19)(23,33)(24,34)(25,28)(26,35)(27,31)(30,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16,18,31)(11,20)(13,23)(14,24)(15,27,19,29)(17,30,25,36)(21,33)(22,34)(26,37)(28,38,32,39)(35,40);(1,4,5,3)(2,7,6,8)(9,13,14,26)(10,17)(11,21,22,35)(12,23,24,37)(15,28)(16,30)(18,25)(19,32)(20,33,34,40)(27,38)(29,39)(31,36)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348308);