Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3277 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)(13,14,19,24)(15,25,27,36)(16,30,33,37)(17,22,18,23)(20,28,35,29)(21,32,38,31)(26,40,41,34)(39,42,44,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,28)(18,29)(19,27)(21,34)(22,25)(23,36)(24,35)(30,44)(31,42)(32,43)(33,41)(37,39)(38,40);(6,9)(10,12)(13,16,19,33)(14,21,24,38)(15,26,27,41)(17,31,18,32)(20,34,35,40)(22,37,23,30)(25,39,36,44)(28,42,29,43);(5,7)(6,10)(8,11)(9,12)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,31,33,32)(20,25,35,36)(21,37,38,30)(26,42,41,43)(34,39,40,44)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440772);