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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1752 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,26)(21,27)(22,37)(23,25)(24,31)(29,39)(30,45)(32,48)(33,43)(34,41)(35,40)(36,44)(38,47)(42,46);(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,29,21,35)(18,24,27,28,25,40,37,39)(20,32,33,46,34,30,36,47)(26,42,44,48,41,38,43,45);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,25)(19,29)(26,41)(27,37)(30,32)(31,35)(43,44)(46,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,23,34)(18,26,25,41)(19,30,29,32)(21,36,22,33)(24,38,40,42)(27,44,37,43)(28,45,39,48)(31,47,35,46);(17,21,23,22)(18,27,25,37)(19,31,29,35)(20,33,34,36)(24,39,40,28)(26,43,41,44)(30,46,32,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366078);