Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1833 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)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(23,27)(29,30)(31,37)(32,34)(33,36)(35,38)(39,44)(40,41)(42,43);(5,7,8,11)(6,9,10,12)(16,24)(20,25)(23,27)(26,28)(30,32)(31,38)(33,41)(43,44);(5,7)(6,9)(8,11)(10,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,35)(32,37)(33,42)(34,38)(36,44)(39,41)(40,43);(1,2)(3,4)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,44,42,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,43)(37,44)(38,42)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5120648);