Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #106 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#334 |
| prime radical | C23 | #334 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
174 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,28,31,42)(20,25)(21,34)(22,37)(23,27)(24,30,39,44)(26,35)(29,41,32,46)(33,40)(36,47)(38,43,48,45);(17,19,21,30,22,32,35,45,23,31,36,44,20,29,33,43)(18,24,26,28,25,38,34,41,27,39,40,42,37,48,47,46);(17,20,23,22)(18,25,27,37)(19,29,31,32)(21,33,36,35)(24,38,39,48)(26,34,40,47)(28,41,42,46)(30,43,44,45)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227192, #72231924, #72231926, … |
| maximal normal subgroup of | #72227192, #72231924, #72231926, … |
| maximal quotient of | #72227188, C2×#122919, #72286592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30115);