Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,36)(16,21,31,39)(17,22,18,23)(25,33,27,35)(26,34,42,44)(29,37,41,43)(30,38,32,40);(6,9)(10,12)(13,15)(14,20)(16,29)(17,27)(18,25)(19,28)(21,37)(22,35)(23,33)(24,36)(26,30)(31,41)(32,42)(34,38)(39,43)(40,44);(1,2,3,4)(15,25)(17,18)(20,33)(22,23)(26,29)(27,28)(30,32)(34,37)(35,36)(38,40)(41,42)(43,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,34)(22,38)(23,40)(24,39)(25,29)(27,41)(28,42)(33,37)(35,43)(36,44);(13,17,19,18)(14,22,24,23)(15,27,28,25)(16,30,31,32)(20,35,36,33)(21,38,39,40)(26,41,42,29)(34,43,44,37)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350776);