Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23393 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,25)(22,32)(24,39)(26,38)(27,41)(28,36)(29,44)(31,34)(33,40)(35,42)(37,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,21,34)(16,27,28,24)(17,31,32,25)(20,35,36,33)(22,38,30,23)(29,40,43,42)(37,39,44,41);(2,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,36)(22,37)(23,39)(25,40)(26,27)(30,44)(31,42)(32,43)(34,35)(38,41);(5,7,8,11)(6,10,9,12)(14,21)(19,34)(20,36)(22,30)(23,38)(33,35)(37,44)(39,41);(16,28)(20,36)(24,27)(29,43)(33,35)(37,44)(39,41)(40,42)Polycyclic presentation:
2^9|b^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342224);