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 | #28484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(13,14,17,21)(15,23,27,19)(16,20,30,38)(18,22,32,40)(24,35,26,33)(25,36,28,34)(29,37,31,39)(41,44,42,43);(6,10)(11,12)(13,15,17,27)(14,19,21,23)(16,28,30,25)(18,24,32,26)(20,36,38,34)(22,33,40,35)(29,42,31,41)(37,44,39,43);(1,2,3,4)(15,24)(16,29)(18,32)(19,33)(20,37)(22,40)(23,35)(26,27)(30,31)(38,39)(41,42)(43,44);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,34)(21,38)(22,37)(23,36)(24,41)(26,42)(27,28)(31,32)(33,43)(35,44)(39,40);(5,8,9,7)(6,12,10,11)(15,26)(16,29)(18,32)(19,35)(20,37)(22,40)(23,33)(24,27)(25,28)(30,31)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6492169);