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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,25)(16,29)(17,33)(18,35)(19,22)(20,28)(21,31)(23,34)(24,27)(26,40)(30,39)(32,38)(36,44)(37,42)(41,43);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,17,27,34)(14,20,24,25,22,23,35,33)(16,30,32,44,31,26,41,42)(21,37,38,40,29,36,43,39);(6,10)(7,11)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,36)(22,29)(23,37)(24,38)(25,39)(27,41)(28,42)(33,40)(34,44)(35,43);(2,4)(6,10)(9,12)(14,22)(15,17)(16,31)(24,35)(28,34)(32,41)(36,37)(39,40);(13,17,18,34,19,15,27,28)(14,23,24,33,22,20,35,25)(16,26,32,42,31,30,41,44)(21,36,38,39,29,37,43,40)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7434772);