Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,23,20,24,18,17)(15,25,27,37,28,36,34,21)(16,29,31,30,32,22,33,38)(26,40,42,35,43,39,41,44);(5,6)(7,11)(8,10)(9,12)(13,15,20,28)(14,21,24,37)(16,26,32,43)(17,25,23,36)(18,27,19,34)(22,35,29,44)(30,39,38,40)(31,41,33,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,30)(18,33)(19,31)(20,32)(21,35)(23,38)(24,29)(25,39)(27,42)(28,43)(34,41)(36,40)(37,44);(7,12)(9,11)(13,17,18,24,20,23,19,14)(15,21,34,36,28,37,27,25)(16,30,33,29,32,38,31,22)(26,35,41,40,43,44,42,39);(6,10)(9,11)(13,18,20,19)(14,23,24,17)(15,27,28,34)(16,31,32,33)(21,36,37,25)(22,30,29,38)(26,41,43,42)(35,39,44,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914126);