Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622904 |
| 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 ◁ C22×C4 ◁ C22×#41 ◁ 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)(3,13)(4,10)(5,18)(6,9)(7,12)(8,16)(11,20)(14,29)(15,28)(17,26)(19,25)(21,30)(22,27)(23,32)(24,31);(1,3)(2,8)(4,14)(5,19)(6,21)(7,15)(9,23)(10,24)(11,28)(12,25)(13,18)(16,20)(17,32)(22,31)(26,29)(27,30);(2,9)(4,6)(8,23)(10,12)(11,27)(13,29)(14,21)(17,22)(18,26)(24,25)(28,30)(31,32);(1,4,7,6)(2,10,12,9)(3,14,15,21)(5,17,20,22)(8,24,25,23)(11,27,18,26)(13,29,28,30)(16,31,19,32);(3,15)(8,25)(13,28)(14,21)(16,19)(23,24)(29,30)(31,32)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234207, #72234211, #72261053, … |
| maximal normal subgroup of | #72234207, #72234211, #72261053, … |
| maximal quotient of | #72238411, #72286439, #72286789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530101);