Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1754 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)(7,11)(10,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,25)(26,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(6,9)(10,12)(14,19)(15,21)(22,27)(23,25)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,20,19,16)(17,25,26,23)(22,24,27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,24,26,28)(22,25,27,23)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4510296);