Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1476 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)(4,8)(5,7)(17,18,22,26,23,27,21,35)(19,28,31,38,29,40,37,24)(20,25,33,43,34,44,36,42)(30,45,47,41,32,48,46,39);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,32)(21,31)(22,37)(23,29)(25,41)(26,40)(27,38)(28,35)(30,34)(33,47)(36,46)(39,44)(42,48)(43,45);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(19,29)(24,38)(28,40)(30,32)(31,37)(39,41)(45,48)(46,47);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,20)(18,25)(19,30)(21,36)(22,33)(23,34)(24,39)(26,43)(27,44)(28,45)(29,32)(31,47)(35,42)(37,46)(38,41)(40,48);(17,21,23,22)(18,26,27,35)(19,31,29,37)(20,33,34,36)(24,40,38,28)(25,42,44,43)(30,46,32,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368994);