Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,23)(18,35)(19,25)(20,24)(21,30)(27,38)(28,37)(29,39)(31,44)(32,43)(33,41)(34,42)(36,40);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,31)(17,33)(18,29)(19,30)(20,36)(22,41)(23,44)(24,39)(25,26)(27,32)(28,34)(35,40)(37,43)(38,42);(1,2,4,3)(6,10)(9,12)(13,16,19,17)(14,22,25,23)(15,27,30,28)(18,32,20,34)(21,37,26,38)(24,42,35,43)(29,33,36,31)(39,41,40,44);(1,3,4,2)(13,17,19,16)(14,23,25,22)(15,28,30,27)(18,34,20,32)(21,38,26,37)(24,43,35,42)(29,31,36,33)(39,44,40,41);(13,18,19,20)(14,24,25,35)(15,29,30,36)(16,32,17,34)(21,39,26,40)(22,42,23,43)(27,33,28,31)(37,41,38,44)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348819);