Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622875 |
| 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×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3899 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,24)(13,25)(15,21)(16,22)(17,30)(18,23)(19,27)(20,28)(26,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,26)(16,27)(17,29)(18,28)(21,31)(22,24)(23,25)(30,32);(1,4)(2,8)(3,12)(5,17)(6,15)(7,19)(9,22)(10,21)(11,24)(13,26)(14,29)(16,18)(20,31)(23,30)(25,32)(27,28);(4,15)(8,21)(12,26)(16,17)(19,31)(22,30)(24,32)(27,29);(3,13)(7,20)(11,25)(12,26)(14,28)(19,31)(24,32)(27,29)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,h^a=hi,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233205, #72234840, #72235130, … |
| maximal normal subgroup of | #72233205, #72234840, #72235130, … |
| maximal quotient of | #72232683, #72233116, #72233180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530071);