Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1670 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)(5,6,9,7)(8,10,12,11)(13,14,18,19)(15,20,22,17)(16,24,27,21)(23,26,25,28)(30,34)(31,37)(36,39)(38,43);(14,19)(17,20)(21,24)(26,28)(29,30,33,34)(31,35,37,41)(32,39,40,36)(38,44,43,42);(6,7)(10,11)(13,15,18,22)(14,20,19,17)(16,25,27,23)(21,26,24,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(43,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,27)(14,21,19,24)(15,23,22,25)(17,26,20,28)(29,31,33,37)(30,35,34,41)(32,38,40,43)(36,42,39,44);(5,8,9,12)(6,10,7,11)(13,17,18,20)(14,22,19,15)(16,26,27,28)(21,25,24,23)(29,32)(30,36)(31,38)(33,40)(34,39)(35,42)(37,43)(41,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843822);