Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C2×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,25)(16,30)(17,22)(18,35)(19,24)(20,28)(21,32)(23,26)(27,39)(29,38)(31,44)(33,36)(34,41)(37,42)(40,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,26,42,19,29,18,28)(14,20,35,38,24,37,23,25)(16,17,27,43,31,33,32,34)(21,36,44,40,39,22,30,41);(7,11)(9,12)(13,16,18,32,19,31,26,27)(14,21,23,30,24,39,35,44)(15,17,28,34,29,33,42,43)(20,36,25,41,37,22,38,40);(7,11)(9,12)(13,17)(14,22)(15,27)(16,28)(18,34)(19,33)(20,30)(21,38)(23,40)(24,36)(25,39)(26,43)(29,32)(31,42)(35,41)(37,44);(16,31)(17,33)(21,39)(22,36)(27,32)(30,44)(34,43)(40,41)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364560);