Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1867 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)(5,6,10,8)(7,11,9,12)(13,14,17,18)(15,19,23,21)(16,24,25,20)(22,27,28,26)(29,30)(31,32)(33,38)(34,35)(36,37)(39,40)(41,44)(42,43);(5,7,10,9)(6,11,8,12)(14,18)(19,21)(20,24)(26,27)(30,35)(31,37)(38,43)(39,44);(5,8,10,6)(7,11,9,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,31)(30,36)(32,35)(33,39)(34,37)(38,41)(40,43)(42,44);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,32,34,36)(30,37,35,31)(33,41,42,40)(38,39,43,44);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33,34,42)(30,38,35,43)(31,39,37,44)(32,40,36,41)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2670540);