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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1800 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)(8,9)(10,12)(13,14,19,24)(15,26,29,21)(16,30,34,42)(17,36,28,38)(18,25,20,23)(22,35,41,33)(27,43,44,39)(31,40,32,37);(2,4)(6,9)(10,11)(13,15,19,29)(14,21,24,26)(16,31,34,32)(17,18,28,20)(22,39,41,43)(23,38,25,36)(27,35,44,33)(30,37,42,40);(13,16,20,35,19,34,18,33)(14,22,25,42,24,41,23,30)(15,27,28,32,29,44,17,31)(21,37,38,43,26,40,36,39);(1,2,3,4)(13,17,20,15,19,28,18,29)(14,21,25,38,24,26,23,36)(16,32,35,44,34,31,33,27)(22,40,42,39,41,37,30,43);(5,7,8,12)(6,10,9,11)(13,18,19,20)(14,23,24,25)(15,28,29,17)(16,33,34,35)(21,38,26,36)(22,30,41,42)(27,32,44,31)(37,43,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=f,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, 7521156);