Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,19,24)(15,25,29,35)(16,21,33,40)(17,22,18,23)(20,28,36,26)(27,37,42,43)(30,34,41,44)(31,38,32,39);(2,4)(13,15,19,29)(14,20,24,36)(16,30,33,41)(17,28,18,26)(21,37,40,43)(22,25,23,35)(27,32,42,31)(34,38,44,39);(5,7,8,12)(6,9,10,11)(15,26)(17,18)(20,25)(22,23)(27,30)(28,29)(31,32)(34,37)(35,36)(38,39)(41,42)(43,44);(13,16)(14,21)(15,27)(17,31)(18,32)(19,33)(20,34)(22,38)(23,39)(24,40)(25,37)(26,30)(28,41)(29,42)(35,43)(36,44);(13,17,19,18)(14,22,24,23)(15,26,29,28)(16,31,33,32)(20,35,36,25)(21,38,40,39)(27,30,42,41)(34,43,44,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,e^c=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, 6352707);