Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,25)(19,27,30,23)(20,32,33,24)(22,26,35,39)(28,36,43,41)(29,37,31,40)(34,45,46,38)(42,48,44,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30)(18,23,25,27)(20,28,33,43)(22,29,35,31)(24,36,32,41)(26,40,39,37)(34,42,46,44)(38,48,45,47);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20)(18,24)(19,28)(21,33)(22,34)(23,36)(25,32)(26,38)(27,41)(29,42)(30,43)(31,44)(35,46)(37,47)(39,45)(40,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(18,25)(19,29)(22,35)(23,37)(24,32)(27,40)(28,42)(30,31)(34,46)(36,47)(41,48)(43,44);(1,4,5,3)(2,7,8,6)(18,25)(20,33)(23,27)(26,39)(28,43)(34,46)(37,40)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7460900);