Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2666 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,26,30,21)(16,20,29,37)(17,22,35,41)(18,23,27,24)(28,43,32,38)(31,36,44,39)(33,40,34,42);(1,2)(3,4)(6,9)(10,12)(13,15,27,29,19,30,18,16)(14,20,23,21,25,37,24,26)(17,32,33,31,35,28,34,44)(22,39,42,43,41,36,40,38);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,30,19,29,27,15)(14,21,24,20,25,26,23,37)(17,31,34,32,35,44,33,28)(22,38,40,36,41,43,42,39);(2,4)(13,17)(14,22)(15,28)(16,31)(18,34)(19,35)(20,36)(21,38)(23,42)(24,40)(25,41)(26,43)(27,33)(29,44)(30,32)(37,39);(13,18,19,27)(14,23,25,24)(15,29,30,16)(17,33,35,34)(20,26,37,21)(22,40,41,42)(28,31,32,44)(36,38,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348025);