Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15693 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
755 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)(3,11,14,7)(4,15,17,8)(5,18)(6,10)(9,19)(12,22)(13,24,27,21)(16,23,29,28)(20,26)(25,31)(30,32);(1,3)(2,7)(4,12,17,26)(5,13)(6,14)(8,22,15,20)(9,24)(10,11)(16,25,29,32)(18,21)(19,27)(23,30,28,31);(1,4)(2,8)(3,12,14,26)(5,16)(6,17)(7,20,11,22)(9,23)(10,15)(13,25,27,32)(18,28)(19,29)(21,30,24,31);(1,5)(2,9,10,18)(3,13)(4,16)(6,19)(7,21,11,24)(8,23,15,28)(12,25)(14,27)(17,29)(20,30,22,31)(26,32)Polycyclic presentation:
2^9|a^2=eg,b^2=fg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237790, #72237793, #72237802, … |
| maximal normal subgroup of | #72237790, #72237793, #72237802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348943);