Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3303 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1831 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)(7,10)(8,11)(9,12)(13,14,18,23,19,24,32,34)(15,25,27,22,28,38,17,20)(16,21,31,37,30,36,42,43)(26,39,41,35,40,44,29,33);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,20,24,22)(16,26,30,40)(17,32,27,18)(21,33,36,35)(23,38,34,25)(29,42,41,31)(37,44,43,39);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,21)(15,26)(17,29)(18,31)(19,30)(20,33)(22,35)(23,37)(24,36)(25,39)(27,41)(28,40)(32,42)(34,43)(38,44);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,27,32,28)(14,22,23,38,24,20,34,25)(16,29,31,26,30,41,42,40)(21,35,37,44,36,33,43,39);(5,9,10,8)(6,11,7,12)(16,30)(21,36)(26,40)(29,41)(31,42)(33,35)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106156);