Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433173 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2336 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)(3,12)(4,9)(5,7)(6,10)(8,13)(11,18)(14,22)(15,21)(16,20)(17,23)(19,27)(24,29)(25,26)(28,30)(31,32);(2,7)(3,13)(5,18)(8,21)(9,20)(10,11)(12,19)(14,26)(16,29)(22,31)(23,24)(25,30);(1,3,5,15,6,13,18,27)(2,8,11,21,10,12,7,19)(4,14,16,28,17,26,29,32)(9,22,24,31,23,25,20,30);(1,4)(2,9)(3,14)(5,16)(6,17)(7,20)(8,22)(10,23)(11,24)(12,25)(13,26)(15,28)(18,29)(19,30)(21,31)(27,32);(2,10)(7,11)(8,12)(9,23)(19,21)(20,24)(22,25)(30,31)Polycyclic presentation:
2^9|c^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72318474, #72318482, #72318509, … |
| maximal normal subgroup of | #72318474, #72318482, #72318509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340369);