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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
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)(35,36)(37,45)(38,39)(40,41)(42,46)(43,48)(44,47);(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)(35,41)(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)(34,40)(36,39)(37,47)(38,41)(42,44)(43,46)(45,48);(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)(33,36,38,40)(34,41,39,35)(37,48,46,44)(42,43,45,47);(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)(34,42,39,45)(35,43,41,47)(36,44,40,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fh,d^2=h,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, 4097421);