Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,15)(14,18)(17,22)(20,23)(21,24);(10,15)(11,20)(12,21)(13,22)(17,18)(23,24);(2,5)(6,8)(9,11)(10,16)(12,14)(13,21)(15,23)(17,24)(18,19)(20,22);(9,12)(10,16)(11,14)(13,20)(15,23)(17,24)(18,19)(21,22);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307949, C2×#6453874 |
| maximal normal subgroup of | #72307949, C2×#6453874 |
| maximal quotient of | #72308044, C2×#6453874, #72309326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361070);