Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,9,11,8,10,12,7)(13,14,17,21,18,23,15,22,19,24,25,20,27,28,16,26);(1,2,3,4)(6,10)(7,11)(14,20)(15,25)(16,17)(18,27)(21,24)(22,23)(26,28);(5,7,12,10,8,11,9,6)(13,15,18,25,19,16,27,17)(14,21,28,26,24,20,23,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,c^b=ch,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318455, #72318470, #72318493, … |
| maximal normal subgroup of | #72318455, #72318470, #72318493, … |
| maximal quotient of | #72322460, #72322473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36373);