Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14335 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3526 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)(4,8)(5,18)(6,10)(7,14)(9,19)(12,26)(13,25)(15,29)(16,31)(17,24)(20,28)(21,27)(22,30)(23,32);(7,20)(11,25)(21,22)(26,29);(1,3)(2,7,10,20)(4,15,17,12)(5,14)(6,13)(8,22)(9,11,18,25)(16,30,32,27)(19,28)(21,24)(23,29)(26,31);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,24)(11,26)(13,15)(14,27)(18,31)(19,32)(20,22)(25,29)(28,30)Polycyclic presentation:
2^9|c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216192, #72216294, … |
| maximal normal subgroup of | #72216047, #72216192, #72216294, … |
| maximal quotient of | #72237193, C2×#382440, #72255932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289636);