Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
431 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,22)(19,26)(20,27)(23,28)(24,29)(25,32)(30,31);(17,18,23,32,20,25,19,22,21,24,26,31,27,30,28,29);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,24)(22,29)(25,30)(31,32);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^9|b^2=egi,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321876 |
| maximal normal subgroup of | #72321876 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398335);