Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,27,38)(16,28,30,42)(17,31,25,21)(19,32,34,24)(20,35,36,29)(26,40,41,33)(37,43,44,39);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(15,24)(16,29)(17,25)(19,23)(20,28)(21,31)(26,41)(27,32)(30,35)(33,40)(34,38)(36,42);(6,9)(10,11)(13,15,25,32,18,27,17,24)(14,19,31,38,22,34,21,23)(16,26,35,43,30,41,29,39)(20,33,42,44,36,40,28,37);(5,7,8,12)(6,10,9,11)(13,16,17,29,18,30,25,35)(14,20,21,28,22,36,31,42)(15,26,24,39,27,41,32,43)(19,33,23,37,34,40,38,44);(1,3,4,2)(16,30)(20,36)(26,41)(28,42)(29,35)(33,40)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503996);