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 | 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,27,17,21)(16,30,33,40)(18,26,20,24)(22,32,41,34)(23,29,38,36)(28,42,35,39)(31,43,44,37);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,31)(17,19)(18,29)(20,36)(22,39)(23,26)(24,38)(25,27)(28,34)(30,37)(32,35)(33,44)(40,43)(41,42);(5,7,8,11)(6,10,9,12)(13,16,18,32,19,33,20,34)(14,22,24,40,25,41,26,30)(15,28,36,44,17,35,29,31)(21,37,23,42,27,43,38,39);(2,4)(13,17,18,29,19,15,20,36)(14,23,24,27,25,38,26,21)(16,28,32,44,33,35,34,31)(22,39,40,37,41,42,30,43);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,29,17,36)(16,32,33,34)(21,38,27,23)(22,40,41,30)(28,31,35,44)(37,39,43,42)Polycyclic presentation:
2^9|a^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=cg,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, 7468118);