Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14,19,35,18,23,17,24)(15,25,34,20,27,21,28,38)(16,29,36,41,33,22,32,42)(26,39,43,44,31,40,30,37);(6,9)(7,11)(13,15,18,27)(14,20,23,38)(16,30,33,43)(17,34,19,28)(21,35,25,24)(22,40,29,39)(26,36,31,32)(37,42,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,28)(18,27)(19,34)(20,24)(22,40)(23,25)(26,33)(29,39)(30,36)(32,43)(35,38)(37,42)(41,44);(6,9)(10,12)(13,16,19,36,18,33,17,32)(14,22,35,42,23,29,24,41)(15,26,34,43,27,31,28,30)(20,37,21,39,38,44,25,40);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(22,29)(24,35)(26,31)(28,34)(30,43)(41,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7052978);