Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2989 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)(13,14,19,32,18,23,17,22)(15,24,26,20,28,34,27,35)(16,21,29,36,30,37,31,38)(25,39,42,33,41,44,40,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,32)(29,40)(30,41)(31,42)(36,39)(37,43)(38,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,30)(19,29)(20,33)(22,38)(23,37)(24,39)(26,42)(27,40)(28,41)(32,36)(34,44)(35,43);(6,9)(10,12)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,29,30,31)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(5,7,8,11)(6,10,9,12)(16,30)(21,37)(25,41)(29,31)(33,43)(36,38)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=fi,d^2=i,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398405);