Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1768 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)(13,14,18,19)(15,16,22,20)(17,21,26,27)(23,24,28,25)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(6,9)(10,12)(14,19)(15,22)(21,27)(23,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,25,27,24)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,25,26,24)(21,28,27,23)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2671504);