Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14913 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(5,18)(8,12)(9,20)(10,21)(13,29)(14,27)(15,16)(23,24)(26,30)(31,32);(1,2,5,7,6,10,18,21)(3,12,27,19,14,8,11,25)(4,9,15,20,17,24,16,23)(13,28,29,31,30,22,26,32);(1,3,5,11,6,14,18,27)(2,8,7,19,10,12,21,25)(4,13,15,26,17,30,16,29)(9,22,20,31,24,28,23,32);(1,4)(2,9)(3,13)(5,15)(6,17)(7,20)(8,22)(10,24)(11,26)(12,28)(14,30)(16,18)(19,31)(21,23)(25,32)(27,29)Polycyclic presentation:
2^9|b^2=eh,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240445, #72240458, … |
| maximal normal subgroup of | #72237692, #72240445, #72240458, … |
| maximal quotient of | #72237700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312692);