Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2374 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,23)(18,24)(19,21)(20,28)(22,34)(26,41)(27,37)(30,44)(31,40)(32,38)(33,35)(36,43)(39,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,30)(17,33)(18,28)(19,27)(21,35)(22,39)(23,37)(24,25)(26,32)(29,41)(31,42)(34,43)(36,40)(38,44);(2,4)(5,8,9,12)(6,10,7,11)(14,21)(17,19)(20,35)(22,38)(23,24)(25,37)(27,33)(29,40)(31,34)(36,41)(39,44)(42,43);(6,7)(10,11)(13,16)(14,22)(15,26)(17,31)(18,32)(19,34)(20,36)(21,38)(23,29)(24,40)(25,39)(27,42)(28,30)(33,43)(35,41)(37,44);(13,17,18,19)(14,23,24,21)(15,27,28,33)(16,31,32,34)(20,35,25,37)(22,29,40,38)(26,42,30,43)(36,41,39,44)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408993);