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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2061 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,29)(17,23)(18,22)(19,24)(21,31)(25,37)(26,35)(27,40)(28,36)(30,34)(32,38)(33,39)(41,43)(42,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,28)(14,20,24,36)(16,30,32,42)(17,27,18,26)(21,37,39,43)(22,35,23,40)(25,33,41,31)(29,34,38,44);(13,16)(14,21)(15,25)(17,31)(18,33)(19,32)(20,34)(22,29)(23,38)(24,39)(26,30)(27,42)(28,41)(35,43)(36,44)(37,40);(2,4)(5,7,8,11)(6,10,9,12)(14,22)(15,26)(17,18)(21,29)(23,24)(25,30)(27,28)(31,33)(35,40)(37,43)(38,39)(41,42);(6,9)(10,12)(13,17,19,18)(14,23,24,22)(15,26,28,27)(16,31,32,33)(20,35,36,40)(21,38,39,29)(25,30,41,42)(34,43,44,37)Polycyclic presentation:
2^9|b^2=i,d^2=f,e^2=i,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=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 7423929);