Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,31)(24,28)(25,38)(26,29)(27,33)(30,39)(32,45)(34,46)(35,50)(36,43)(37,52)(40,48)(41,49)(42,51)(44,47);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,25,33,26,34,39,49)(22,27,30,31,29,41,38,46)(24,35,37,51,36,32,47,48)(28,42,44,50,43,40,52,45);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,32)(25,37)(26,36)(27,40)(29,43)(30,44)(31,45)(33,48)(34,35)(38,52)(39,47)(41,42)(46,50)(49,51);(6,11)(10,16)(12,18)(17,20)(22,29)(24,36)(27,41)(30,38)(31,46)(32,35)(37,47)(48,51);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,36)(28,43)(32,35)(37,47)(40,42)(44,52)(45,50)(48,51)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344440);