Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23857 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,19,24)(15,20,29,37)(16,22,31,38)(17,23,18,33)(21,32,39,30)(25,40,26,35)(27,36,28,43)(34,42,44,41);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,29)(21,34)(22,40)(23,36)(24,37)(26,31)(30,41)(32,42)(33,43)(35,38)(39,44);(2,4)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,34)(22,33)(23,38)(24,39)(26,29)(27,41)(28,42)(35,36)(37,44)(40,43);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,35)(21,33)(23,39)(24,38)(25,29)(27,41)(28,42)(34,36)(37,40)(43,44);(2,4)(6,9)(10,12)(13,17,19,18)(14,23,24,33)(15,27,29,28)(16,30,31,32)(20,36,37,43)(21,38,39,22)(25,41,26,42)(34,35,44,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579129);