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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 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)(15,27,31,21)(16,32,34,39)(17,36,33,43)(18,26,20,24)(22,29,42,35)(23,37,41,28)(30,40,38,44);(2,4)(6,9)(10,12)(13,15,19,31)(14,21,25,27)(16,33,34,17)(18,30,20,38)(22,41,42,23)(24,40,26,44)(28,29,37,35)(32,36,39,43);(5,7,8,11)(6,10,9,12)(13,16,18,29,19,34,20,35)(14,22,24,39,25,42,26,32)(15,28,38,17,31,37,30,33)(21,36,44,23,27,43,40,41);(1,2,3,4)(13,17,18,37,19,33,20,28)(14,23,24,43,25,41,26,36)(15,29,38,34,31,35,30,16)(21,39,44,42,27,32,40,22);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,30,31,38)(16,29,34,35)(17,28,33,37)(21,40,27,44)(22,39,42,32)(23,36,41,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,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, 7520924);