Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,23)(16,27)(17,31)(18,21)(19,26)(20,29)(22,32)(24,39)(25,38)(28,37)(30,36)(33,44)(34,42)(35,41)(40,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,26,18,25,32,42)(14,19,22,23,21,34,31,38)(16,28,30,44,29,24,40,41)(20,35,36,39,27,33,43,37);(2,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,30)(18,29)(19,33)(21,27)(22,36)(23,37)(25,28)(26,41)(31,43)(32,40)(34,35)(38,39)(42,44);(5,7,8,11)(6,10,9,12)(14,21)(16,29)(19,34)(22,31)(23,38)(24,28)(30,40)(41,44);(15,25)(19,34)(23,38)(24,28)(26,42)(33,35)(37,39)(41,44)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342744);