Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,20,32)(12,16)(13,25,27,17)(14,18)(15,22)(21,33,34,40)(23,36,38,30)(24,31)(26,28)(29,35)(37,39);(9,11)(10,15)(12,21)(13,26)(14,20)(16,29)(17,32)(18,28)(19,25)(22,27)(23,37)(24,34)(30,40)(31,39)(33,36)(35,38);(1,3,5,4)(2,6,8,7)(11,20)(15,28)(19,32)(21,34)(22,26)(29,39)(33,40)(35,37);(9,12,14,24)(10,16,18,31)(11,21,20,34)(13,23,27,38)(15,29,28,39)(17,30,25,36)(19,33,32,40)(22,35,26,37)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fgi,f^e=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72293174, #72293175, #72293179, … |
| maximal normal subgroup of | #72293174, #72293175, #72293179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387305);