Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15376 |
| 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 | |
1354 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)(4,14,16,28)(5,10)(6,9)(8,17,25,20)(11,29,15,32)(12,21)(13,23)(18,27)(19,26)(22,30,24,31);(1,3)(2,7)(4,15)(5,18)(6,12)(8,24)(9,21)(10,27)(11,16)(13,19)(14,29)(17,30)(20,31)(22,25)(23,26)(28,32);(5,19)(7,21)(10,26)(13,18)(14,28)(17,20)(22,24)(30,31);(1,4)(2,8)(3,11,12,15)(5,20)(6,16)(7,22,21,24)(9,25)(10,28)(13,31,18,30)(14,26)(17,19)(23,32,27,29)Polycyclic presentation:
2^9|a^2=e,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319162, #72319438, #72319465, … |
| maximal normal subgroup of | #72319162, #72319438, #72319465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335129);