Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34)(35,36)(37,45)(38,39)(40,42)(41,48)(43,47)(44,46);(2,6)(7,8)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(24,26)(28,29)(34,39)(35,42)(41,45)(43,46);(1,2,5,6)(3,7,4,8)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,35)(34,40)(36,39)(37,46)(38,42)(41,44)(43,48)(45,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,36,38,40)(34,35,39,42)(37,47,48,44)(41,46,45,43);(1,4,5,3)(2,8,6,7)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,41)(35,43)(36,44)(38,48)(39,45)(40,47)(42,46)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=gh,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3301860);