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) = C25×C4 |
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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10,14,12)(11,15,13,16)(18,22)(19,25)(20,27)(30,32)(33,34)(35,38)(36,41)(37,42)(39,44)(40,46)(43,47)(45,48);(9,11,14,13)(10,15,12,16)(18,22)(19,25)(24,29)(26,31)(34,35)(39,41)(40,42)(47,48);(1,2,4,6)(3,5,7,8)(9,12,14,10)(11,15,13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,41,44,39)(37,40,46,42)(43,47,45,48);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,15,12,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,44)(34,39,35,41)(37,45,46,43)(40,48,42,47);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,40,35,42)(36,43,44,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=fgi,e^2=f,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, 3438805);