Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,8,7)(11,15)(13,16)(18,22)(19,25)(20,28)(30,32)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,47,48);(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,15,16)(18,22)(20,28)(23,26)(27,31)(34,39)(35,42)(37,46)(44,48);(1,3,5,8)(2,6,4,7)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,42)(34,36,39,41)(37,47,46,43)(40,48,45,44);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36)(34,35)(37,48)(38,41)(39,42)(40,47)(43,45)(44,46);(1,4,5,2)(3,7,8,6)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,46)(34,40,39,45)(35,43,42,47)(36,44,41,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=h,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4097428);