Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394273 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×B(4,𝔽2) | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,13)(14,20);(2,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)Polycyclic presentation:
2^9|b^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,e^c=ei,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225628, #72225632, #72225636, … |
| maximal normal subgroup of | #72225628, #72225632, #72225636, … |
| maximal quotient of | #72233079, #72233131, #72234594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312470);