Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,35)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,42,44)(25,34,30,36)(33,40,43,41);(2,3)(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,20,23,35)(16,24,29,42)(17,30,18,25)(21,33,38,43)(22,36,31,34)(27,39,37,44)(28,41,32,40);(1,2,4,3)(7,12)(9,11)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,42)(27,31)(30,41)(34,39)(35,43)(36,44);(6,10)(9,11)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,35,36)(21,37,38,27)(24,40,42,41)(33,39,43,44);(1,3,4,2)(14,23)(20,35)(21,38)(22,31)(27,37)(33,43)(34,36)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fh,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717570);