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 |
| prime radical | #42882 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,18,23,19,24,17,32)(15,20,27,38,28,39,33,37)(16,29,30,41,31,22,34,21)(25,36,43,40,26,44,42,35);(1,2)(3,4)(6,9)(7,11)(13,15,19,28)(14,20,24,39)(16,26,31,25)(17,33,18,27)(21,40,41,35)(22,36,29,44)(23,38,32,37)(30,42,34,43);(5,6)(7,10)(8,9)(11,12)(13,16,19,31)(14,21,24,41)(15,25,28,26)(17,30,18,34)(20,35,39,40)(22,32,29,23)(27,42,33,43)(36,38,44,37);(2,4)(6,9)(10,12)(13,16)(14,22)(15,26)(17,34)(18,30)(19,31)(20,36)(21,23)(24,29)(25,28)(27,42)(32,41)(33,43)(35,37)(38,40)(39,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,32)(15,27,28,33)(16,30,31,34)(20,37,39,38)(21,22,41,29)(25,42,26,43)(35,36,40,44)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055423);