Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,20,26)(15,27,30,22)(16,31,29,21)(17,23,33,38)(18,25,19,24)(28,41,42,37)(32,44,43,36)(34,40,35,39);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16,20,30,18,29)(14,21,25,22,26,31,24,27)(17,28,35,32,33,42,34,43)(23,36,40,37,38,44,39,41);(6,9)(7,11)(13,16,19,30,20,29,18,15)(14,22,25,31,26,27,24,21)(17,32,35,42,33,43,34,28)(23,37,40,44,38,41,39,36);(5,7,8,11)(6,10,9,12)(13,17,20,33)(14,23,26,38)(15,28,30,42)(16,32,29,43)(18,34,19,35)(21,36,31,44)(22,37,27,41)(24,39,25,40);(6,9)(10,12)(17,33)(23,38)(28,42)(32,43)(34,35)(36,44)(37,41)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427666);