Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2171 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)(8,12)(9,10)(13,14,20,24)(15,21,27,38)(16,28,33,44)(17,34,26,37)(18,23,19,35)(22,31,41,32)(25,42,43,39)(29,40,30,36);(1,2)(3,4)(6,10)(11,12)(13,15)(14,21)(16,29)(17,19)(18,26)(20,27)(22,39)(23,37)(24,38)(25,31)(28,40)(30,33)(32,43)(34,35)(36,44)(41,42);(5,7,9,8)(6,11,10,12)(13,16,19,32,20,33,18,31)(14,22,23,28,24,41,35,44)(15,25,26,30,27,43,17,29)(21,36,34,42,38,40,37,39);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,27,20,26,19,15)(14,21,35,37,24,38,23,34)(16,30,31,25,33,29,32,43)(22,40,44,42,41,36,28,39);(6,10)(11,12)(13,18,20,19)(14,23,24,35)(15,26,27,17)(16,31,33,32)(21,37,38,34)(22,28,41,44)(25,30,43,29)(36,39,40,42)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fgi,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469462);