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×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14,18,34,20,25,19,24)(15,26,35,21,29,38,28,39)(16,30,33,22,17,23,32,41)(27,36,42,37,31,40,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,27)(18,28)(19,35)(20,29)(22,40)(23,37)(24,38)(25,39)(26,34)(30,44)(32,43)(33,42)(36,41);(6,9)(10,12)(13,16,20,17)(14,22,25,41)(15,27,29,31)(18,32,19,33)(21,36,39,40)(23,34,30,24)(26,37,38,44)(28,42,35,43);(6,9)(10,12)(13,17,20,16)(14,23,25,30)(15,27,29,31)(18,32,19,33)(21,37,39,44)(22,34,41,24)(26,36,38,40)(28,43,35,42);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,25,34)(15,28,29,35)(16,32,17,33)(21,38,39,26)(22,23,41,30)(27,42,31,43)(36,44,40,37)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^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=dhi,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, 6398748);