Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#3301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1911 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)(5,6,9,11)(7,12,8,10)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,34)(25,35)(26,36)(27,37)(28,38)(29,33)(30,40)(31,39)(41,43)(42,44);(5,6)(7,10)(8,12)(9,11)(13,15,18,25)(14,19,21,35)(16,26,27,42)(17,29,31,24)(20,36,37,44)(22,33,39,34)(23,30,41,28)(32,40,43,38);(1,2,3,4)(13,16)(14,20)(15,23)(17,30)(18,27)(19,32)(21,37)(22,40)(24,42)(25,41)(26,29)(28,31)(33,36)(34,44)(35,43)(38,39);(5,7,9,8)(6,10,11,12)(16,27)(20,37)(23,41)(26,42)(28,30)(32,43)(36,44)(38,40);(5,8,9,7)(6,10,11,12)(14,21)(15,24)(17,31)(19,33)(20,37)(23,42)(25,29)(26,41)(28,30)(32,36)(34,35)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593990);