Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,21)(12,26)(13,16)(14,19)(15,20)(17,24)(18,29)(22,34)(23,38)(25,31)(27,35)(28,36)(30,32)(33,39)(37,40);(1,2,4,5)(3,6,7,8)(10,16)(11,22)(13,14)(18,32)(19,20)(21,31)(23,37)(24,25)(26,36)(27,39)(28,29)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,27)(13,24)(14,22)(15,25)(16,31)(18,35)(19,21)(20,34)(23,28)(26,38)(29,39)(30,37)(32,33)(36,40);(3,7)(6,8)(9,12,15,30)(10,18,20,36)(11,23,25,39)(13,29,14,28)(16,32,19,26)(17,33,34,38)(21,35,31,40)(22,37,24,27)Polycyclic presentation:
2^9|b^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362907);