Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1784 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)(5,6,10,8)(7,11,9,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,34,35)(31,36,40,43)(32,33,42,37)(38,44,41,39);(5,7,10,9)(6,11,8,12)(14,18)(16,24)(19,21)(22,28)(30,35)(33,37)(36,43)(39,44);(5,8,10,6)(7,11,9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,31)(30,36)(32,41)(33,39)(34,40)(35,43)(37,44)(38,42);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,14)(15,21)(16,25)(17,18)(19,23)(20,24)(22,26)(27,28)(29,32,34,42)(30,37,35,33)(31,38,40,41)(36,39,43,44);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,37)(30,32,35,42)(31,39,40,44)(36,41,43,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2984636);