Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1943 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)(5,6,8,10)(7,9,11,12)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(5,7,8,11)(6,9,10,12)(14,16)(19,21)(20,24)(26,27)(31,35)(32,38)(39,43)(40,44);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2509368);