Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3027 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3027 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,18)(13,20);(5,6)(7,13,14,20)(8,12)(9,11)(10,15,18,16)(17,19);(5,7)(6,10)(8,16)(9,14)(11,18)(12,20)(13,19)(15,17);(2,4)(8,17)(12,19)(13,20)(15,16)Polycyclic presentation:
2^9|b^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233807, #72239252, #72256724, … |
| maximal normal subgroup of | #72233807, #72239252, #72256724, … |
| maximal quotient of | #72233805, #72239263, C2×#417819 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325015);