Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1865 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)(13,14,18,23)(15,20,27,38)(16,29,30,21)(17,32,31,22)(19,24,34,41)(25,40,33,36)(26,44,43,37)(28,39,35,42);(6,10)(11,12)(13,15,18,27)(14,20,23,38)(16,26,30,43)(17,33,31,25)(19,35,34,28)(21,37,29,44)(22,40,32,36)(24,42,41,39);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,21,23,29)(15,25,27,33)(17,34,31,19)(20,36,38,40)(22,41,32,24)(26,28,43,35)(37,39,44,42);(5,8,9,7)(6,12,10,11)(16,17)(19,34)(21,22)(24,41)(25,43)(26,33)(28,35)(29,32)(30,31)(36,44)(37,40)(39,42);(13,17,18,31)(14,22,23,32)(15,26,27,43)(16,19,30,34)(20,37,38,44)(21,24,29,41)(25,35,33,28)(36,42,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491733);