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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,25)(23,32)(26,35)(27,36)(28,42)(29,37)(30,38)(31,39)(33,41)(34,40)(43,44);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,31)(17,26)(18,28)(19,30)(21,39)(22,35)(23,37)(24,42)(25,38)(27,33)(29,32)(34,43)(36,41)(40,44);(5,7,8,12)(6,10,9,11)(13,16,32,34,19,33,18,17)(14,21,23,40,25,41,24,22)(15,26,28,27,30,43,29,31)(20,35,42,36,38,44,37,39);(13,17,18,33,19,34,32,16)(14,22,24,41,25,40,23,21)(15,27,29,26,30,31,28,43)(20,36,37,35,38,39,42,44);(13,18,19,32)(14,23,25,24)(15,28,30,29)(16,17,33,34)(20,37,38,42)(21,40,41,22)(26,27,43,31)(35,39,44,36)Polycyclic presentation:
2^9|b^2=fg,c^2=ghi,d^2=h,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, 6373692);