Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,25,17,23)(15,20,28,36,29,37,32,41)(16,22,30,38,31,40,33,21)(26,39,43,44,27,35,42,34);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,25,37)(16,26,31,27)(17,32,18,28)(21,34,38,44)(22,39,40,35)(23,41,24,36)(30,43,33,42);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,26)(17,30)(18,33)(19,31)(20,34)(22,23)(24,40)(25,38)(27,29)(28,42)(32,43)(35,36)(37,44)(39,41);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,27)(17,33)(18,30)(19,31)(20,35)(21,23)(24,38)(25,40)(26,29)(28,42)(32,43)(34,36)(37,39)(41,44);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,28,29,32)(16,30,31,33)(20,36,37,41)(21,22,38,40)(26,42,27,43)(34,35,44,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971604);