Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2006 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)(13,14,18,22)(15,24,27,20)(16,29,21,38)(17,32,34,31)(19,36,35,23)(25,44,39,33)(26,30,43,41)(28,40,37,42);(1,2)(3,4)(6,9)(10,12)(13,15,18,27)(14,20,22,24)(16,30,21,41)(17,33,34,44)(19,37,35,28)(23,42,36,40)(25,32,39,31)(26,38,43,29);(13,16,18,21)(14,17,22,34)(15,25,27,39)(19,32,35,31)(20,26,24,43)(23,38,36,29)(28,41,37,30)(33,40,44,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,26)(16,22)(18,34)(19,38)(20,39)(23,31)(24,25)(27,43)(28,44)(29,35)(30,40)(32,36)(33,37)(41,42);(6,9)(10,12)(14,22)(17,34)(20,24)(23,36)(26,43)(29,38)(33,44)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,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,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423822);