Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232766 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2011 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 2011 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22);(2,7)(3,4)(6,8)(15,23)(18,24);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#233610, #72213800, #72213848, … |
| maximal normal subgroup of | C2×#233610, #72213800, #72213848, … |
| maximal quotient of | #72211654, #72211656, C2×#233610, … |
| automorphism group of | #1313 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140806);