Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2154 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)(5,6,8,9)(7,11,12,10)(13,14)(15,21)(16,30)(17,23)(18,34)(19,25)(20,24)(22,31)(26,43)(27,37)(28,40)(29,39)(32,42)(33,36)(35,38)(41,44);(1,2,3,4)(6,9)(10,11)(13,15,19,29)(14,21,25,39)(16,27,31,44)(17,33,32,26)(18,28,20,35)(22,41,30,37)(23,36,42,43)(24,38,34,40);(5,7,8,12)(6,10,9,11)(13,16,20,32,19,31,18,17)(14,22,24,23,25,30,34,42)(15,26,28,27,29,33,35,44)(21,36,40,41,39,43,38,37);(13,17,18,31,19,32,20,16)(14,23,34,22,25,42,24,30)(15,27,35,26,29,44,28,33)(21,37,38,43,39,41,40,36);(13,18,19,20)(14,24,25,34)(15,28,29,35)(16,17,31,32)(21,38,39,40)(22,23,30,42)(26,27,33,44)(36,37,43,41)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=ghi,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374005);