Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13867 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,10)(3,13,17,8,15,23,16,21)(4,9,18,24,19,25,20,26)(14,28,30,32,29,27,31,22);(8,21)(13,23)(22,32)(27,28);(1,3)(2,8)(4,14)(5,17)(6,16)(7,15)(9,22)(10,23)(11,21)(12,13)(18,31)(19,29)(20,30)(24,27)(25,32)(26,28);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,22)(10,26)(11,25)(12,24)(13,27)(15,29)(16,31)(17,30)(21,32)(23,28)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216059, #72216181, #72216294, … |
| maximal normal subgroup of | #72216059, #72216181, #72216294, … |
| maximal quotient of | C2×#359562, #72249730, #72255890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266758);