Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2022 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)(13,14,18,19)(15,22,20,16)(17,25,28,21)(23,24,26,27)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(6,9)(10,12)(14,19)(15,20)(21,25)(23,26)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,6,8,9)(7,10,11,12)(13,15,18,20)(14,16,19,22)(17,26,28,23)(21,27,25,24)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,19)(17,27)(18,22)(21,23)(24,28)(25,26)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(13,17)(14,21)(15,23)(16,24)(18,28)(19,25)(20,26)(22,27)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3252408);