Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,4,3)(6,10)(11,12)(13,14,19,20)(15,24,28,21)(16,17,22,31)(18,33,34,23)(25,26,29,35)(27,36,41,38)(30,43,37,32)(39,44,42,40);(5,6,9,10)(7,11,8,12)(14,20)(17,31)(21,24)(23,33)(26,35)(32,43)(36,38)(40,44);(5,7,9,8)(6,11,10,12)(13,15,19,28)(14,21,20,24)(16,29,22,25)(17,26,31,35)(18,27,34,41)(23,36,33,38)(30,42,37,39)(32,40,43,44);(13,16,19,22)(14,17,20,31)(15,25,28,29)(18,30,34,37)(21,35,24,26)(23,32,33,43)(27,39,41,42)(36,44,38,40);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,26)(16,20)(18,32)(19,31)(21,25)(23,37)(24,29)(27,40)(28,35)(30,33)(34,43)(36,39)(38,42)(41,44);(13,18,19,34)(14,23,20,33)(15,27,28,41)(16,30,22,37)(17,32,31,43)(21,36,24,38)(25,39,29,42)(26,40,35,44)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532966);