Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #29816 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14,19,23)(15,20,27,37)(16,21,31,40)(17,22,32,18)(24,35,42,43)(25,26,36,33)(28,38,41,44)(29,30,39,34);(5,6)(7,10)(8,12)(9,11)(13,15,19,27)(14,20,23,37)(16,28,31,41)(17,25,32,36)(18,33,22,26)(21,38,40,44)(24,34,42,30)(29,43,39,35);(2,4)(13,16,19,31)(14,21,23,40)(15,24,27,42)(17,29,32,39)(18,34,22,30)(20,35,37,43)(25,38,36,44)(26,41,33,28);(5,7,9,8)(6,10,11,12)(13,17)(14,22)(15,25)(16,29)(18,23)(19,32)(20,26)(21,30)(24,38)(27,36)(28,43)(31,39)(33,37)(34,40)(35,41)(42,44);(5,8,9,7)(6,10,11,12)(15,26)(17,32)(18,22)(20,36)(24,41)(25,37)(27,33)(28,42)(29,39)(30,34)(35,44)(38,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583452);