Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,20,24,19,26,18,25)(15,27,30,21,32,42,31,38)(16,22,35,40,34,41,17,23)(28,39,29,36,43,37,33,44);(6,9)(10,12)(13,15)(14,21)(16,33)(17,28)(18,30)(19,32)(20,31)(22,39)(23,36)(24,27)(25,42)(26,38)(29,34)(35,43)(37,41)(40,44);(1,2,3,4)(13,16,20,35,19,34,18,17)(14,22,24,40,26,41,25,23)(15,28,30,29,32,43,31,33)(21,36,42,37,38,44,27,39);(5,7,8,11)(6,10,9,12)(13,17,18,34,19,35,20,16)(14,23,25,41,26,40,24,22)(15,29,31,28,32,33,30,43)(21,37,27,36,38,39,42,44);(13,18,19,20)(14,24,26,25)(15,30,32,31)(16,17,34,35)(21,27,38,42)(22,40,41,23)(28,29,43,33)(36,39,44,37)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367188);