Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,18,23,19,24,16,21)(15,25,26,34,28,38,32,20)(17,22,31,37,30,36,29,35)(27,39,40,43,41,44,42,33);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,24,34)(16,26,18,32)(17,27,30,41)(21,25,23,38)(22,33,36,43)(29,40,31,42)(35,39,37,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,18)(14,21,24,23)(15,26,28,32)(17,29,30,31)(20,25,34,38)(22,35,36,37)(27,40,41,42)(33,39,43,44);(7,12)(9,11)(13,17)(14,22)(15,27)(16,29)(18,31)(19,30)(20,33)(21,35)(23,37)(24,36)(25,39)(26,40)(28,41)(32,42)(34,43)(38,44);(6,10)(9,11)(17,30)(22,36)(27,41)(29,31)(33,43)(35,37)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407385);