Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,8)(7,11,9,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,28)(22,23)(25,27)(29,30,34,35)(31,37,39,33)(32,40,41,36)(38,44,42,43);(5,7,10,9)(6,11,8,12)(14,17)(19,22)(20,24)(27,28)(30,35)(32,41)(33,37)(38,42);(5,8,10,6)(7,11,9,12)(13,15,18,23)(14,19,17,22)(16,25,26,21)(20,28,24,27)(29,31)(30,33)(32,42)(34,39)(35,37)(36,43)(38,41)(40,44);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,16,18,26)(14,20,17,24)(15,21,23,25)(19,27,22,28)(29,32,34,41)(30,36,35,40)(31,38,39,42)(33,44,37,43);(13,17,18,14)(15,22,23,19)(16,20,26,24)(21,27,25,28)(29,33)(30,31)(32,43)(34,37)(35,39)(36,42)(38,40)(41,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgh,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3044357);