Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | C2×#3285 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,27,36)(16,21,28,37)(17,22,32,41)(18,23,26,35)(25,34,30,39)(29,38,43,44)(31,40,33,42);(1,2,4,3)(6,9)(10,12)(15,17)(16,28)(18,26)(20,22)(21,37)(23,35)(25,43)(27,32)(29,30)(34,44)(36,41)(38,39);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,29)(17,26)(18,32)(19,27)(21,38)(22,35)(23,41)(24,36)(25,33)(28,43)(30,31)(34,42)(37,44)(39,40);(13,16,19,28)(14,21,24,37)(15,25,27,30)(17,29,32,43)(18,33,26,31)(20,34,36,39)(22,38,41,44)(23,42,35,40);(13,17,19,32)(14,22,24,41)(15,26,27,18)(16,30,28,25)(20,35,36,23)(21,39,37,34)(29,31,43,33)(38,40,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486483);