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 | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,21,32)(12,16)(13,25,27,17)(14,18)(15,22)(20,33,34,40)(23,36,38,30)(24,31)(26,29)(28,35)(37,39);(9,11)(10,15)(12,20)(13,26)(14,21)(16,28)(17,32)(18,29)(19,25)(22,27)(23,37)(24,34)(30,40)(31,39)(33,36)(35,38);(1,3,5,4)(2,6,8,7)(9,12,14,24)(10,16,18,31)(11,20,21,34)(13,23,27,38)(15,28,29,39)(17,30,25,36)(19,33,32,40)(22,35,26,37);(11,21)(15,29)(19,32)(20,34)(22,26)(28,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,c^2=hi,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^e=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72266599 |
| maximal normal subgroup of | #72266599 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387307);