Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,4,7)(2,5,6,8)(11,15)(13,16)(19,25)(23,28)(26,31)(30,32)(34,35)(39,41)(40,42)(46,47);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fgi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557158);