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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1696 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)(7,10,12,11)(14,18)(15,21)(16,24)(26,28)(29,30)(31,32)(33,41)(34,35)(36,37)(38,42)(39,44)(40,43);(5,6,9,8)(7,11,12,10)(16,24)(20,25)(23,27)(26,28)(30,35)(31,37)(33,42)(40,44);(5,7,9,12)(6,10,8,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,36)(32,35)(33,43)(34,37)(38,40)(39,42)(41,44);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,36)(30,37,35,31)(33,44,42,40)(38,39,41,43);(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,42)(30,38,35,41)(31,39,37,43)(32,40,36,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=gh,d^2=fgh,e^2=h,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, 4069060);