Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C16 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1712 |
| commutator, Frattini, cosocle | C82 | C23 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(18,22)(19,23)(20,26)(24,32)(25,30)(27,31)(28,29);(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6);(17,18,23,29,20,24,27,30,21,25,31,32,26,28,19,22)Polycyclic presentation:
2^9|b^2=df,c^2=eg,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi134 conjugacy classes — view the full character table.
| maximal subgroup of | #72213956, #72214187, C2×#123403, … |
| maximal normal subgroup of | #72213956, #72214187, C2×#123403, … |
| maximal quotient of | C2×#123403, #72216263, #72216358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30599);