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×#3309 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2039 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)(13,14,18,19)(15,23,26,20)(16,21,29,33)(17,30,32,22)(24,27,40,34)(25,35,42,38)(28,37,31,36)(39,44,41,43);(6,9)(7,11)(13,15)(14,20)(16,27)(17,25)(18,26)(19,23)(21,24)(22,35)(28,43)(29,34)(30,38)(31,44)(32,42)(33,40)(36,39)(37,41);(2,4)(13,16,18,29)(14,21,19,33)(15,24,26,40)(17,31,32,28)(20,34,23,27)(22,37,30,36)(25,41,42,39)(35,43,38,44);(6,9)(10,12)(13,17,18,32)(14,22,19,30)(15,25,26,42)(16,28,29,31)(20,35,23,38)(21,36,33,37)(24,39,40,41)(27,43,34,44);(5,7,8,11)(6,10,9,12)(14,19)(15,20)(16,29)(22,30)(23,26)(24,27)(25,35)(28,31)(34,40)(38,42)(39,43)(41,44)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,e^a=ei,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, 6580025);