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) = 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 | |
1618 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)(5,6,9,7)(8,10,12,11)(13,14)(15,20)(16,25)(17,21)(18,19)(22,23)(24,27)(26,28)(30,32)(31,37)(33,41)(43,44);(5,7,9,6)(8,10,12,11)(14,19)(16,21)(20,27)(22,28)(29,30,34,32)(31,38,37,35)(33,36,41,40)(39,44,42,43);(5,8)(6,10)(7,11)(9,12)(13,15)(14,20)(16,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,31,34,37)(30,35,32,38)(33,42,41,39)(36,44,40,43);(1,2,3,4)(13,16)(14,17)(15,22)(18,21)(19,25)(20,26)(23,27)(24,28)(29,32,34,30)(31,38,37,35)(33,36,41,40)(39,43,42,44);(13,17)(14,21)(15,23)(16,19)(18,25)(20,22)(24,26)(27,28)(29,33,34,41)(30,36,32,40)(31,39,37,42)(35,43,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=fgi,e^2=g,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, 3373073);