Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,9,10,16)(4,13,14,8)(6,11,15,12)(17,18,21,24)(19,22,26,31)(20,23,28,25)(27,29,32,30);(1,3,5,10)(2,6,7,15)(4,11,14,12)(8,16,13,9)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(3,10)(6,15)(9,16)(11,12)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(3,11)(4,14)(6,16)(8,13)(9,15)(10,12);(2,7)(3,10)(8,13)(11,12)(18,24)(19,26)(23,25)(27,32)Polycyclic presentation:
2^9|a^2=hi,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286361, #72306764, #72306794, … |
| maximal normal subgroup of | #72286361, #72306764, #72306794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274427);