Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 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)(13,14,19,25)(15,22,29,21)(16,30,28,38)(17,32,35,41)(18,26,20,24)(23,34,42,36)(27,43,44,40)(31,37,33,39);(1,2)(3,4)(6,9)(10,12)(13,15,19,29)(14,21,25,22)(16,18,28,20)(17,33,35,31)(23,40,42,43)(24,38,26,30)(27,34,44,36)(32,37,41,39);(5,7,8,11)(6,10,9,12)(13,16,18,29,19,28,20,15)(14,22,24,38,25,21,26,30)(17,33,34,27,35,31,36,44)(23,37,41,43,42,39,32,40);(2,4)(13,17,18,34,19,35,20,36)(14,23,24,41,25,42,26,32)(15,27,16,31,29,44,28,33)(21,37,30,43,22,39,38,40);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,28,29,16)(17,34,35,36)(21,38,22,30)(23,41,42,32)(27,33,44,31)(37,40,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=gi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468414);