Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,19,20,37)(12,23,27,38)(13,28,25,18)(15,30,31,21)(17,34,33,26)(22,39,29,35)(24,36,40,32);(1,2)(3,6)(4,5)(7,8)(11,20)(15,31)(19,37)(21,30)(22,29)(24,40)(32,36)(35,39);(9,11,14,20)(10,15,16,31)(12,24,27,40)(13,29,25,22)(17,35,33,39)(18,36,28,32)(19,38,37,23)(21,34,30,26);(2,5)(6,8)(10,16)(15,31)(17,33)(18,28)(19,37)(23,38)(32,36)(35,39);(1,3,4,7)(2,6,5,8)(10,16)(15,31)(17,33)(18,28)(19,37)(23,38)(32,36)(35,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72216909, #72216945, … |
| maximal normal subgroup of | #72216688, #72216909, #72216945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531211);