Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,31)(17,23)(18,24)(19,25)(20,26)(22,28)(27,42)(29,37)(30,38)(32,44)(33,41)(34,40)(35,39)(36,43);(2,4)(13,15,19,30)(14,21,25,38)(16,32,28,35)(17,34,27,33)(18,29,20,36)(22,39,31,44)(23,40,42,41)(24,37,26,43);(6,9)(10,12)(13,16,19,28)(14,22,25,31)(15,27,30,17)(18,34,20,33)(21,23,38,42)(24,41,26,40)(29,35,36,32)(37,44,43,39);(13,17,19,27)(14,23,25,42)(15,28,30,16)(18,35,20,32)(21,22,38,31)(24,39,26,44)(29,34,36,33)(37,40,43,41);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,26)(15,29,30,36)(16,33,28,34)(17,35,27,32)(21,37,38,43)(22,40,31,41)(23,39,42,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361683);