Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4224 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)(8,16)(11,20)(12,17)(15,18);(5,6)(7,11)(8,17)(9,13)(10,15)(12,16)(14,20)(18,19);(2,4)(5,7)(6,10)(8,18)(9,14)(11,17)(12,20)(13,19)(15,16);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,20)(16,18)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72213913, #72215601, … |
| maximal normal subgroup of | #72213603, #72213913, #72215601, … |
| maximal quotient of | #72210545, #72213205, #72213763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301272);