Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,19,22)(15,24,27,35)(16,21,28,38)(17,23,18,31)(20,32,34,26)(25,39,40,44)(29,36,30,37)(33,41,43,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,39)(37,44)(38,43);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,30)(18,29)(19,28)(20,33)(22,38)(23,37)(24,39)(26,42)(27,40)(31,36)(32,41)(34,43)(35,44);(5,7,8,11)(6,10,9,12)(14,22)(16,28)(20,34)(23,31)(24,35)(25,40)(29,30)(41,42);(6,9)(10,12)(13,17,19,18)(14,23,22,31)(15,26,27,32)(16,29,28,30)(20,35,34,24)(21,36,38,37)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426962);