Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23929 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3127 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)(7,11)(10,12)(13,14,19,24)(15,25,29,40)(16,21,31,38)(17,22,32,39)(18,33,30,23)(20,34,37,28)(26,36,42,44)(27,41,43,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,27)(18,34)(19,29)(21,35)(22,36)(23,40)(24,37)(25,33)(28,30)(31,42)(32,43)(38,41)(39,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,18)(19,31)(20,35)(22,33)(23,39)(24,38)(25,36)(27,34)(28,43)(29,42)(30,32)(37,41)(40,44);(6,9)(10,12)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,36)(21,23)(24,39)(25,41)(26,28)(29,43)(33,38)(34,42)(35,40)(37,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,38)(22,39)(26,42)(27,43)(35,41)(36,44)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398124);