Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,26,30)(21,24,27,28)(25,29,32,31);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(18,22)(19,25)(21,28)(23,31)(24,27)(26,32)(29,30);(10,14)(13,15)(17,19)(18,23)(20,26)(21,29)(22,30)(24,32)(25,28)(27,31);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,18,20,22)(19,23,26,30)(21,24,27,28)(25,29,32,31)Polycyclic presentation:
2^9|a^2=ef,b^2=efi,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72320553, C2×#511530 |
| maximal normal subgroup of | #72320553, C2×#511530 |
| maximal quotient of | #72321384, C2×#511530 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418726);