Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #24364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,28)(17,23)(18,21)(19,24)(22,30)(25,36)(26,38)(27,34)(29,35)(31,39)(32,37)(33,40)(41,43)(42,44);(1,2)(3,4)(6,9)(10,11)(13,15)(14,20)(16,29)(17,25)(18,32)(19,27)(21,37)(22,38)(23,36)(24,34)(26,30)(28,35)(31,41)(33,42)(39,43)(40,44);(14,21)(15,25)(17,18)(20,34)(22,28)(23,24)(26,41)(27,32)(29,42)(30,33)(35,43)(39,40);(2,4)(13,16,19,31)(14,22,24,40)(15,26,27,42)(17,30,18,33)(20,35,34,43)(21,28,23,39)(25,41,32,29)(36,38,37,44);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,23,24,21)(15,25,27,32)(16,30,31,33)(20,36,34,37)(22,39,40,28)(26,41,42,29)(35,38,43,44)Polycyclic presentation:
2^9|d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425883);