Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1822 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)(5,6,8,11)(7,9,12,10)(13,14,18,19)(15,16,20,24)(17,21,26,27)(22,23,28,25)(29,30)(31,36)(32,38)(33,35)(34,37)(39,43)(40,44)(41,42);(5,7)(6,9)(8,12)(10,11)(14,19)(15,20)(21,27)(22,28)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(5,7,8,12)(6,10,11,9)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,19)(17,25)(18,24)(21,22)(23,26)(27,28)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(5,6)(7,10)(8,11)(9,12)(13,17)(14,21)(15,22)(16,23)(18,26)(19,27)(20,28)(24,25)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524653);