Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2938 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)(3,4)(7,11)(10,12)(13,14,20,34,19,25,18,24)(15,26,33,21,30,40,29,37)(16,22,17,23,31,38,32,39)(27,36,43,35,42,44,28,41);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,28)(18,33)(19,30)(20,29)(22,35)(23,36)(24,40)(25,37)(26,34)(31,42)(32,43)(38,41)(39,44);(2,4)(6,9)(10,12)(13,16)(14,22)(15,27)(17,20)(18,32)(19,31)(21,35)(23,34)(24,39)(25,38)(26,36)(28,29)(30,42)(33,43)(37,41)(40,44);(6,9)(10,12)(13,17)(14,23)(15,28)(16,18)(19,32)(20,31)(21,36)(22,24)(25,39)(26,41)(27,33)(29,42)(30,43)(34,38)(35,40)(37,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(22,38)(23,39)(27,42)(28,43)(35,41)(36,44)Polycyclic presentation:
2^9|a^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398455);