Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,27)(16,31)(17,23)(18,24)(19,25)(20,26)(21,29)(22,30)(28,39)(32,44)(33,37)(34,40)(35,41)(36,42)(38,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,29,20,30,19,16)(14,21,25,27,26,31,24,22)(17,28,35,33,36,43,34,32)(23,37,40,39,42,44,41,38);(6,9)(7,11)(13,16,19,30,20,29,18,15)(14,22,24,31,26,27,25,21)(17,33,34,28,36,32,35,43)(23,39,41,37,42,38,40,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,28)(16,32)(18,35)(19,34)(20,36)(21,37)(22,38)(24,41)(25,40)(26,42)(27,39)(29,33)(30,43)(31,44);(6,9)(10,12)(13,18,20,19)(14,24,26,25)(15,29,30,16)(17,34,36,35)(21,22,31,27)(23,40,42,41)(28,32,43,33)(37,39,44,38)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427734);