Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| 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 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(14,20)(21,24);(2,6)(3,4)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(1,3,5,4)(2,6,8,7);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257774, #72257897, #72281671, … |
| maximal normal subgroup of | #72257774, #72257897, #72281671, … |
| maximal quotient of | #72257872, #72257958 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253694);