Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 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)(7,8)(9,11)(10,12)(13,14,19,24)(15,20,27,37)(16,28,32,40)(17,23,18,21)(22,30,39,31)(25,41,29,44)(26,34,33,36)(35,43,38,42);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,29)(17,33)(18,26)(19,27)(21,34)(22,38)(23,36)(24,37)(25,32)(28,44)(30,42)(31,43)(35,39)(40,41);(2,4)(14,21)(17,18)(20,34)(22,28)(23,24)(26,33)(30,31)(35,41)(36,37)(38,44)(39,40)(42,43);(6,11)(7,12)(13,16,19,32)(14,22,24,39)(15,25,27,29)(17,31,18,30)(20,35,37,38)(21,28,23,40)(26,43,33,42)(34,41,36,44);(5,8,9,10)(6,7,11,12)(13,17,19,18)(14,23,24,21)(15,26,27,33)(16,30,32,31)(20,34,37,36)(22,28,39,40)(25,42,29,43)(35,44,38,41)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410262);