Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1710 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)(5,6,9,8)(7,10,11,12)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,27,26,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(7,11)(10,12)(14,15)(17,25)(19,21)(23,26)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,7,9,11)(6,10,8,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,8,9,6)(7,12,11,10)(13,17,18,25)(14,20,15,22)(16,23,24,26)(19,27,21,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=gh,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, 2992681);