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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,16)(17,25)(18,19)(20,22)(21,28)(23,27)(24,26)(29,30,33,37)(31,35,39,34)(32,36,42,41)(38,44,40,43);(6,9)(10,12)(14,19)(15,22)(21,25)(23,26)(29,31,33,39)(30,34,37,35)(32,38,42,40)(36,43,41,44);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,28)(25,27)(29,31)(30,35)(32,40)(33,39)(34,37)(36,43)(38,42)(41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,18,20)(14,15,19,22)(17,27,28,24)(21,26,25,23)(29,30)(31,35)(32,41)(33,37)(34,39)(36,42)(38,43)(40,44);(13,17)(14,21)(15,23)(16,24)(18,28)(19,25)(20,27)(22,26)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,41)(39,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=fgh,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, 3252568);