Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1594 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)(3,5)(4,6)(7,8)(17,18,21,26,22,25,36,42)(19,27,30,38,29,43,35,23)(20,31,34,47,33,24,39,41)(28,45,46,37,32,44,48,40);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,32)(21,35)(22,29)(24,40)(25,38)(26,43)(27,42)(28,33)(30,36)(31,37)(34,46)(39,48)(41,44)(45,47);(2,6)(5,8)(10,13)(11,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,31)(26,41)(27,44)(29,32)(30,46)(35,48)(36,39)(38,40)(42,47)(43,45);(9,11,12,15)(10,14,13,16)(18,25)(20,33)(23,38)(26,42)(27,43)(28,32)(34,39)(46,48);(10,13)(14,16)(18,25)(19,29)(20,33)(26,42)(30,35)(34,39)(37,40)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426364);