Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | C2×#3355 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1874 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)(8,12)(9,10)(13,14,18,22)(15,19,26,35)(16,27,30,20)(17,21,31,38)(23,32,25,34)(24,40,41,33)(28,37,29,36)(39,44,42,43);(6,10)(11,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,33)(21,32)(22,35)(25,31)(27,40)(28,42)(29,39)(30,41)(34,38)(36,44)(37,43);(1,2,3,4)(15,23)(16,28)(17,31)(19,32)(20,36)(21,38)(24,41)(25,26)(27,37)(29,30)(33,40)(34,35);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,29)(18,30)(19,33)(21,37)(22,27)(23,39)(25,42)(26,41)(28,31)(32,43)(34,44)(35,40)(36,38);(5,8,9,7)(6,12,10,11)(15,23)(16,29)(17,31)(19,32)(20,37)(21,38)(25,26)(27,36)(28,30)(34,35)(39,42)(43,44)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492101);