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)(15,26)(16,30)(17,35)(18,25)(19,24)(20,23)(21,28)(22,34)(27,42)(29,37)(31,41)(32,40)(33,39)(36,44)(38,43);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,18)(19,28)(20,29)(22,38)(23,35)(24,26)(25,37)(27,34)(30,36)(32,43)(33,44)(39,41)(40,42);(6,9)(10,11)(13,16,20,34,19,33,18,32)(14,22,25,30,24,40,23,39)(15,27,17,31,28,43,29,44)(21,36,35,38,26,41,37,42);(5,7,8,12)(6,10,9,11)(13,17,20,28,19,29,18,15)(14,21,25,35,24,26,23,37)(16,31,34,43,33,44,32,27)(22,36,30,38,40,41,39,42);(1,3,4,2)(13,18,19,20)(14,23,24,25)(15,17,28,29)(16,32,33,34)(21,35,26,37)(22,39,40,30)(27,31,43,44)(36,38,41,42)Polycyclic presentation:
2^9|b^2=fi,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, 7503997);