Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 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)(7,10)(8,9)(11,12)(13,14,19,16)(15,25,28,21)(17,30,32,22)(18,23,20,24)(26,29,38,34)(27,35,42,39)(31,37,33,36)(40,44,41,43);(6,9)(7,11)(13,15)(14,21)(16,25)(17,27)(18,29)(19,28)(20,34)(22,35)(23,26)(24,38)(30,39)(31,43)(32,42)(33,44)(36,40)(37,41);(2,4)(13,16,19,14)(15,26,28,38)(17,22,32,30)(18,23,20,24)(21,34,25,29)(27,41,42,40)(31,37,33,36)(35,43,39,44);(6,9)(10,12)(13,17,19,32)(14,22,16,30)(15,27,28,42)(18,31,20,33)(21,35,25,39)(23,36,24,37)(26,40,38,41)(29,43,34,44);(5,7,8,11)(6,10,9,12)(13,18)(14,23)(15,28)(16,24)(17,31)(19,20)(21,25)(22,36)(27,42)(30,37)(32,33)(35,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579977);