Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1658 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)(3,8)(4,6)(5,7)(9,10)(11,19,21,15)(12,23,26,16)(13,18)(14,17)(20,30)(22,35,27,31)(24,32)(25,36,28,33)(29,34)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,24)(13,27)(14,21)(16,32)(17,19)(18,35)(20,25)(22,29)(23,40)(26,39)(28,38)(30,33)(31,34)(36,37);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,23)(18,36)(19,37)(21,38)(22,39)(24,27)(25,29)(31,40)(32,35)(33,34);(11,21)(15,19)(20,38)(22,27)(24,39)(30,37)(31,35)(32,40)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265784);