Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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×#3301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1909 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)(7,11)(8,12)(9,10)(13,14,18,23)(15,25,28,20)(16,21,30,38)(17,22,31,39)(19,24,33,41)(26,40,32,35)(27,43,44,36)(29,42,34,37);(6,10)(11,12)(13,15)(14,20)(16,27)(17,32)(18,28)(19,34)(21,36)(22,40)(23,25)(24,42)(26,31)(29,33)(30,44)(35,39)(37,41)(38,43);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,21,23,38)(15,26,28,32)(17,33,31,19)(20,35,25,40)(22,41,39,24)(27,29,44,34)(36,37,43,42);(5,8,9,7)(6,12,10,11)(16,17)(19,33)(21,22)(24,41)(26,44)(27,32)(29,34)(30,31)(35,43)(36,40)(37,42)(38,39);(13,17,18,31)(14,22,23,39)(15,27,28,44)(16,19,30,33)(20,36,25,43)(21,24,38,41)(26,34,32,29)(35,42,40,37)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491721);