Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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 | #35150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3309 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)(5,6)(7,12)(8,10)(9,11)(13,14,20,26)(15,27,32,38)(16,22,35,41)(17,23,34,40)(18,25,19,24)(21,30,39,31)(28,37,43,44)(29,42,33,36);(1,2,3,4)(13,15,18,31,20,32,19,30)(14,21,24,27,26,39,25,38)(16,33,17,28,35,29,34,43)(22,37,40,36,41,44,23,42);(5,7,9,8)(6,10,11,12)(13,16,20,35)(14,22,26,41)(15,28,32,43)(17,18,34,19)(21,36,39,42)(23,25,40,24)(27,37,38,44)(29,31,33,30);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,29)(16,18)(19,35)(20,34)(21,37)(22,25)(24,41)(26,40)(27,42)(28,31)(30,43)(32,33)(36,38)(39,44);(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,35,17)(21,38,39,27)(22,23,41,40)(28,29,43,33)(36,44,42,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359379);