Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622907 |
| 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 | |
3756 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)(3,9)(4,7)(5,8)(6,12)(10,16)(11,18)(13,24)(14,21)(15,23)(17,20)(19,22)(25,27)(26,28)(29,31)(30,32);(2,7)(4,16)(5,18)(8,20)(9,21)(10,12)(11,22)(14,27)(15,28)(23,31)(24,25)(26,32);(2,8)(5,18)(7,20)(9,23)(10,22)(11,12)(15,28)(17,19)(21,31)(24,26)(25,32)(29,30);(1,3)(2,9)(4,14)(5,15)(6,13)(7,21)(8,23)(10,25)(11,26)(12,24)(16,27)(17,29)(18,28)(19,30)(20,31)(22,32);(3,13)(9,24)(14,27)(15,28)(21,25)(23,26)(29,30)(31,32)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72260900, #72261053, #72261130 |
| maximal normal subgroup of | #72260900, #72261053, #72261130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530103);