Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361803 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13903 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,16,18,8)(5,19)(6,9)(10,20)(12,21,15,23)(13,22)(17,24,30,29)(25,27)(26,32)(28,31);(1,3,5,15,6,14,20,12)(2,7,19,21,9,11,10,23)(4,13,17,28,18,27,30,26)(8,22,29,31,16,25,24,32);(3,12)(5,20)(7,21)(10,19)(11,23)(13,26)(14,15)(17,30)(22,31)(24,29)(25,32)(27,28);(1,4)(2,8)(3,13)(5,17)(6,18)(7,22)(9,16)(10,24)(11,25)(12,26)(14,27)(15,28)(19,29)(20,30)(21,31)(23,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72240437, #72240442, #72240457, … |
| maximal normal subgroup of | #72240437, #72240442, #72240457, … |
| maximal quotient of | #72255602, #72255969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268999);