Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,10,8,12)(13,14,18,25,19,26,20,24)(15,27,31,39,32,40,30,21)(16,33,17,23,34,22,35,42)(28,41,44,37,36,38,29,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,32)(14,21,26,39)(16,29,34,44)(17,36,35,28)(18,30,20,31)(22,41,33,38)(23,37,42,43)(24,27,25,40);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,17,19,34,20,35)(14,22,25,42,26,33,24,23)(15,28,31,44,32,36,30,29)(21,37,27,38,39,43,40,41);(2,4)(13,17,18,34,19,35,20,16)(14,23,25,22,26,42,24,33)(15,29,31,28,32,44,30,36)(21,38,27,43,39,41,40,37);(13,18,19,20)(14,24,26,25)(15,30,32,31)(16,17,34,35)(21,27,39,40)(22,23,33,42)(28,29,36,44)(37,38,43,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fgi,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=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355313);