Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1825 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)(7,12)(8,10)(9,11)(13,14,19,23)(15,24,28,40)(16,29,32,38)(17,22,18,33)(20,27,36,25)(21,34,39,30)(26,35,31,37)(41,42,44,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,25)(16,30)(17,18)(20,24)(21,29)(22,33)(27,28)(32,34)(36,40)(38,39)(41,44)(42,43);(13,15,19,28)(14,20,23,36)(16,31,32,26)(17,25,18,27)(21,37,39,35)(22,40,33,24)(29,44,38,41)(30,43,34,42);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,26)(17,30)(18,34)(19,32)(20,35)(22,38)(23,39)(24,41)(25,42)(27,43)(28,31)(29,33)(36,37)(40,44);(6,11)(10,12)(13,17,19,18)(14,22,23,33)(15,27,28,25)(16,30,32,34)(20,24,36,40)(21,38,39,29)(26,43,31,42)(35,41,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^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,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425317);