Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,12)(7,10,8,11)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30)(31,35)(32,39)(33,41)(34,37)(36,42)(38,40)(43,44);(5,7,9,8)(6,10,12,11)(14,15)(19,21)(20,22)(27,28)(29,31,34,39)(30,35,37,32)(33,38,42,44)(36,43,41,40);(5,6)(7,10)(8,11)(9,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,32)(33,38)(34,39)(35,37)(36,40)(41,43)(42,44);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26)(29,32)(30,31)(33,40)(34,35)(36,38)(37,39)(41,44)(42,43);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)(29,33,34,42)(30,36,37,41)(31,38,39,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=f,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342355);