Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,27)(17,32)(18,34)(19,24)(21,31)(22,26)(23,29)(25,37)(28,35)(30,43)(33,36)(38,44)(39,40)(41,42);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,24,22)(16,28,30,44)(17,18,33,29)(21,37,39,42)(23,32,34,36)(25,40,41,31)(27,35,43,38);(6,9)(10,11)(13,16,29,40,19,30,18,31)(14,21,34,43,24,39,23,27)(15,25,33,44,26,41,17,28)(20,35,32,42,22,38,36,37);(5,7,8,12)(6,10,9,11)(13,17,18,26,19,33,29,15)(14,22,23,32,24,20,34,36)(16,28,31,41,30,44,40,25)(21,38,27,42,39,35,43,37);(1,3,4,2)(16,30)(21,39)(25,41)(27,43)(28,44)(31,40)(35,38)(37,42)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=g,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504269);