Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C23 | #443 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,11,13,24,15,25,14,20)(7,16)(12,26,27,34,29,23,28,35)(18,31)(19,32)(33,36);(1,2,3,4)(5,6)(7,12)(8,20)(9,13)(10,15)(11,17)(14,21)(16,29)(18,34)(19,27)(22,24)(23,33)(25,30)(26,36)(28,32)(31,35);(5,7,17,33,9,19,22,31,10,16,30,36,21,32,8,18)(6,12,11,23,13,27,24,35,15,29,25,26,14,28,20,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gi,h^b=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216211, #72216271, #72216414, … |
| maximal normal subgroup of | #72216211, #72216271, #72216414, … |
| maximal quotient of | #72225377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36339);