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 | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14)(15,18)(16,19)(17,26)(20,23)(21,25)(22,24)(27,28);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(14,18)(19,20)(21,27)(22,26);(2,4)(5,8,9,10)(6,11,12,7)(14,19)(16,25)(17,24)(18,20)(21,22)(23,28)(26,27);(6,12)(7,11)(14,20)(16,25)(17,24)(18,19)(21,26)(22,27)(23,28)Polycyclic presentation:
2^9|a^2=ei,b^2=ei,c^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72344000 |
| maximal normal subgroup of | #72344000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416774);