Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,12,11)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27)(29,30,34,35)(31,38,41,36)(32,33,43,37)(39,44,42,40);(5,7,9,12)(6,10,8,11)(16,25)(19,24)(22,28)(26,27)(30,35)(33,37)(36,38)(40,44);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,31)(30,36)(32,42)(33,44)(34,41)(35,38)(37,40)(39,43);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28)(29,32,34,43)(30,37,35,33)(31,39,41,42)(36,44,38,40);(5,8,9,6)(7,11,12,10)(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)(29,33,34,37)(30,32,35,43)(31,40,41,44)(36,39,38,42)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fg,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4064757);