Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C16 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1712 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #12515 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
260 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)(8,12)(9,16)(11,15)(13,14)(18,22)(19,26)(20,27)(23,28)(24,32)(25,31)(29,30);(1,2,7,14,4,9,11,12,5,8,15,16,10,13,3,6);(2,8)(6,12)(9,13)(14,16)(17,18,23,32,20,24,19,22,21,25,26,30,27,29,28,31)Polycyclic presentation:
2^9|b^2=df,c^2=egi,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=gi,c^b=ch86 conjugacy classes — view the full character table.
| maximal subgroup of | #72213956, #72214196, #72225191, … |
| maximal normal subgroup of | #72213956, #72214196, #72225191, … |
| maximal quotient of | #72216196, C3×#123419, #72226113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30615);