Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3204 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2342 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)(7,9)(8,10)(11,12)(13,14,18,23)(15,24,29,40)(16,21,30,20)(17,32,25,22)(19,33,35,28)(26,42,27,39)(31,36,41,38)(34,43,37,44);(5,7)(6,9)(8,11)(10,12)(13,15,25,33,18,29,17,28)(14,19,32,40,23,35,22,24)(16,26,41,44,30,27,31,43)(20,34,38,42,21,37,36,39);(1,2,3,4)(7,11)(9,12)(13,16,18,30)(14,20,23,21)(15,26,29,27)(17,31,25,41)(19,34,35,37)(22,36,32,38)(24,39,40,42)(28,43,33,44);(7,11)(9,12)(13,16)(14,21)(15,27)(17,31)(18,30)(19,34)(20,23)(22,38)(24,39)(25,41)(26,29)(28,44)(32,36)(33,43)(35,37)(40,42);(6,10)(9,12)(16,30)(20,21)(26,27)(31,41)(34,37)(36,38)(39,42)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380474);