Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499908 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33517 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,18)(14,20)(17,25)(19,26)(21,34)(22,31)(23,29)(24,30)(27,33)(28,35)(32,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,15,20,30)(12,22,27,36)(13,21,26,23)(16,31,33,40)(17,28,32,39)(18,34,19,29)(25,35,38,37);(10,16)(11,21)(12,25)(13,26)(15,28)(17,19)(18,32)(20,33)(22,36)(23,24)(27,38)(29,40)(30,39)(31,34);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407153);