Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14,19,25)(15,23,28,21)(16,30,34,42)(17,36,29,38)(18,26,20,24)(22,35,41,33)(27,43,44,39)(31,40,32,37);(1,2)(3,4)(6,10)(11,12)(13,15,19,28)(14,21,25,23)(16,31,34,32)(17,20,29,18)(22,39,41,43)(24,36,26,38)(27,35,44,33)(30,37,42,40);(5,7,9,8)(6,11,10,12)(13,16,20,35,19,34,18,33)(14,22,26,42,25,41,24,30)(15,27,17,32,28,44,29,31)(21,37,36,43,23,40,38,39);(2,4)(5,7,9,8)(6,11,10,12)(13,17,20,28,19,29,18,15)(14,23,26,38,25,21,24,36)(16,32,35,44,34,31,33,27)(22,40,42,39,41,37,30,43);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,17,28,29)(16,33,34,35)(21,36,23,38)(22,30,41,42)(27,32,44,31)(37,43,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472984);