Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 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)(13,14)(15,22)(16,21)(17,23)(18,24)(19,25)(20,26)(27,38)(28,37)(29,36)(30,35)(31,44)(32,40)(33,41)(34,42)(39,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,28,20,29,18,16)(14,21,24,36,26,37,25,22)(17,27,32,31,34,43,33,30)(23,35,41,39,42,44,40,38);(6,9)(7,11)(13,16,18,29,20,28,19,15)(14,22,25,37,26,36,24,21)(17,31,33,27,34,30,32,43)(23,39,40,35,42,38,41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,30)(18,33)(19,32)(20,34)(21,35)(22,38)(24,41)(25,40)(26,42)(28,31)(29,43)(36,39)(37,44);(6,9)(10,12)(13,18,20,19)(14,24,26,25)(15,16,29,28)(17,32,34,33)(21,36,37,22)(23,40,42,41)(27,31,43,30)(35,38,44,39)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427746);