Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,30,37)(16,21,33,41)(17,23,18,22)(20,28,38,29)(26,36,31,44)(27,35,43,39)(32,42,34,40);(5,6)(7,11)(8,10)(9,12)(13,15,19,30)(14,20,24,38)(16,31,33,26)(17,29,18,28)(21,39,41,35)(22,25,23,37)(27,34,43,32)(36,42,44,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,34)(18,32)(19,33)(20,35)(22,42)(23,40)(24,41)(25,36)(27,29)(28,43)(30,31)(37,44)(38,39);(7,12)(9,11)(13,16)(14,21)(15,27)(17,32)(18,34)(19,33)(20,36)(22,40)(23,42)(24,41)(25,35)(26,29)(28,31)(30,43)(37,39)(38,44);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,28,30,29)(16,32,33,34)(20,37,38,25)(21,40,41,42)(26,27,31,43)(35,36,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901296);