Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15259 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1999 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)(3,7)(4,15,17,8)(5,19)(6,10)(9,14)(11,21)(12,27,16,22)(13,23)(18,26,32,30)(20,24)(25,28,29,31);(1,3)(2,7)(4,16,17,12)(5,14)(6,13)(8,22,15,27)(9,19)(10,23)(11,24)(18,31,32,28)(20,21)(25,26,29,30);(1,4)(2,8)(3,12)(5,18)(6,17)(7,22)(9,25)(10,15)(11,26)(13,16)(14,28)(19,30)(20,31)(21,32)(23,27)(24,29);(1,3)(2,9)(4,12)(5,20)(6,13)(7,19)(8,25)(10,24)(11,23)(14,21)(15,29)(16,17)(18,31)(22,30)(26,27)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72240435, #72240443, #72240448, … |
| maximal normal subgroup of | #72240435, #72240443, #72240448, … |
| maximal quotient of | #72251320, #72256040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325177);