Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14915 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1377 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)(4,15)(5,17)(8,18)(9,24)(10,20)(11,25)(12,14)(16,28)(22,23)(26,27)(29,32);(1,2,5,7,6,10,17,20)(3,8,14,22,13,23,12,18)(4,9,15,19,16,24,28,30)(11,21,25,32,27,31,26,29);(1,3)(2,8)(4,11)(5,12)(6,13)(7,18)(9,21)(10,23)(14,17)(15,26)(16,27)(19,29)(20,22)(24,31)(25,28)(30,32);(1,4,5,15,6,16,17,28)(2,9,7,19,10,24,20,30)(3,11,12,26,13,27,14,25)(8,21,18,29,23,31,22,32)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240442, #72240448, … |
| maximal normal subgroup of | #72240432, #72240442, #72240448, … |
| maximal quotient of | #72251298, C2×#405619, #72255992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312815);