Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #59126 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2120 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2120 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,11)(7,9,12,10)(13,14,17,20)(15,18,16,19);(1,2)(3,4)(6,9)(7,12)(10,11)(14,18)(15,16)(19,20);(2,4)(14,19)(15,16)(18,20);(6,9)(7,12)(10,11)(14,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214865, #72219584, #72282068 |
| maximal normal subgroup of | #72214865, #72219584, #72282068 |
| maximal quotient of | #72214424, #72214756, #72214877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153612);