Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,12,22,25,10,16,23,30,8,14,21,29)(7,13,19,27,20,24,33,34,17,28,31,36,18,26,32,35);(1,2,3,4)(6,12,16,14)(7,17)(13,26,28,24)(15,29,30,25)(18,20)(19,31)(27,34,36,35)(32,33);(5,7,11,20,10,17,8,18)(6,13,12,24,16,28,14,26)(9,19,22,33,23,31,21,32)(15,27,25,34,30,36,29,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279571, #72279591, #72279609, … |
| maximal normal subgroup of | #72279571, #72279591, #72279609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34156);