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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1814 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)(15,24,27,35)(16,28,31,38)(17,22,18,32)(20,33,36,25)(21,30,37,29)(26,34,42,44)(39,40,43,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,25)(16,29)(17,18)(20,24)(21,28)(22,32)(27,33)(30,31)(35,36)(37,38)(39,43)(40,41);(5,8,9,7)(6,10,11,12)(13,15,19,27)(14,20,23,36)(16,26,31,42)(17,25,18,33)(21,34,37,44)(22,35,32,24)(28,39,38,43)(29,40,30,41);(2,4)(13,16,19,31)(14,21,23,37)(15,26,27,42)(17,29,18,30)(20,34,36,44)(22,38,32,28)(24,39,35,43)(25,40,33,41);(6,11)(10,12)(13,17,19,18)(14,22,23,32)(15,25,27,33)(16,30,31,29)(20,35,36,24)(21,28,37,38)(26,41,42,40)(34,39,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=i,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, 7422855);