Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3518 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2937 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)(13,14,17,24,19,25,18,23)(15,22,28,37,29,36,27,20)(16,30,33,21,34,40,32,41)(26,38,42,39,31,35,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,25,37)(16,31,34,26)(17,27,18,28)(21,38,41,35)(22,24,36,23)(30,39,40,44)(32,43,33,42);(6,9)(10,12)(13,16,19,34)(14,21,25,41)(15,26,29,31)(17,32,18,33)(20,35,37,38)(22,39,36,44)(23,40,24,30)(27,42,28,43);(6,9)(10,12)(13,15)(14,22)(16,26)(17,28)(18,27)(19,29)(20,23)(21,39)(24,37)(25,36)(30,38)(31,34)(32,43)(33,42)(35,40)(41,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,32,34,33)(20,22,37,36)(21,40,41,30)(26,42,31,43)(35,39,38,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398706);