Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622911 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,17)(5,11)(6,10)(7,23)(8,14)(9,16)(12,18)(15,28)(19,30)(20,26)(21,25)(22,32)(24,29)(27,31);(1,3,8,7)(2,9,14,17)(4,18,16,6)(5,15,20,22)(10,13,12,23)(11,24,26,30)(19,31,29,21)(25,28,27,32);(1,4,8,16)(2,10,14,12)(3,6,7,18)(5,19,20,29)(9,23,17,13)(11,25,26,27)(15,21,22,31)(24,32,30,28);(1,5)(2,11)(3,15)(4,19)(6,21)(7,22)(8,20)(9,24)(10,25)(12,27)(13,28)(14,26)(16,29)(17,30)(18,31)(23,32);(5,20)(11,26)(15,22)(19,29)(21,31)(24,30)(25,27)(28,32)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530108);