Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,22)(18,34)(19,24)(20,27)(21,26)(23,35)(28,40)(30,38)(31,37)(32,36)(33,42)(39,43)(41,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,28,35,30)(14,20,23,25,24,38,34,40)(16,17,26,33,31,32,43,44)(21,36,29,42,39,22,37,41);(7,11)(9,12)(13,16,18,26,19,31,35,43)(14,21,23,29,24,39,34,37)(15,17,27,33,28,32,30,44)(20,36,25,42,38,22,40,41);(7,11)(9,12)(13,17)(14,22)(15,26)(16,30)(18,33)(19,32)(20,37)(21,25)(23,41)(24,36)(27,31)(28,43)(29,38)(34,42)(35,44)(39,40);(6,10)(9,12)(16,31)(17,32)(21,39)(22,36)(26,43)(29,37)(33,44)(41,42)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380537);