Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1521 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)(20,25,34,40)(21,26,23,28)(30,43,46,37)(31,41,35,38)(33,39,36,42)(44,48,45,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(20,33)(21,23)(25,39)(26,28)(30,44)(31,35)(34,36)(37,47)(38,41)(40,42)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,30)(21,35)(22,32)(23,31)(25,37)(26,41)(27,29)(28,38)(33,44)(34,46)(36,45)(39,47)(40,43)(42,48);(10,14)(13,15)(17,20,22,34)(18,25,27,40)(19,30,32,46)(21,33,23,36)(24,37,29,43)(26,39,28,42)(31,45,35,44)(38,48,41,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,35)(20,33,34,36)(24,38,29,41)(25,39,40,42)(30,45,46,44)(37,48,43,47)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=dgi,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7489133);