Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,36)(20,30,32,45)(21,34,31,25)(24,38,39,33)(27,40,42,48)(29,44,41,37)(35,46,47,43);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,37)(26,36)(30,40)(31,41)(32,42)(33,43)(34,44)(38,46)(39,47)(45,48);(11,15)(13,16)(17,20,31,38,22,32,21,33)(18,24,34,45,26,39,25,30)(19,27,41,46,28,42,29,43)(23,35,44,48,36,47,37,40);(10,14)(13,16)(19,28)(23,36)(27,42)(29,41)(35,47)(37,44)(40,48)(43,46);(1,3,4,7)(2,6,5,8)(20,32)(24,39)(27,42)(30,45)(33,38)(35,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341224);