Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(9,10)(11,15)(12,23)(13,16)(14,19)(17,24)(18,26)(20,34)(21,30)(22,28)(25,33)(27,32)(29,36)(31,37)(35,40)(38,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,28)(17,29)(18,31)(19,30)(23,34)(24,36)(25,35)(26,37)(27,38)(32,39)(33,40);(6,8)(10,16)(13,26)(15,28)(17,23)(18,19)(22,37)(24,27)(29,34)(30,31)(32,33)(36,38)(39,40);(2,4)(9,12,14,25)(10,17,19,32)(11,20,21,35)(13,27,26,24)(15,29,30,39)(16,23,18,33)(22,38,37,36)(28,34,31,40);(11,21)(15,30)(20,35)(22,37)(28,31)(29,39)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408149);