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 | #25578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,29,32,24)(20,33,35,25)(21,26,23,28)(30,38,46,43)(31,41,36,39)(34,42,37,40)(44,47,45,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(20,34)(21,23)(25,40)(26,28)(30,44)(31,36)(33,42)(35,37)(38,47)(39,41)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,41)(27,29)(28,39)(33,43)(34,44)(35,46)(37,45)(40,47)(42,48);(10,14)(13,15)(17,20,22,35)(18,25,27,33)(19,30,32,46)(21,34,23,37)(24,38,29,43)(26,40,28,42)(31,45,36,44)(39,48,41,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,40,33,42)(30,45,46,44)(38,48,43,47)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,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, 7489113);