Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(4,16)(5,7)(9,23)(11,20)(12,27)(13,14)(17,22)(18,19)(21,25)(28,31)(29,30);(1,2)(3,9)(4,17)(5,20)(6,11)(7,8)(10,18)(12,28)(13,21)(14,23)(15,25)(16,26)(19,22)(24,29)(27,32)(30,31);(1,3,7,13,6,15,5,14)(2,9,8,21,11,25,20,23)(4,12,18,27,19,30,16,29)(10,24,22,28,26,32,17,31);(1,4)(2,10)(3,12)(5,16)(6,19)(7,18)(8,22)(9,24)(11,26)(13,27)(14,29)(15,30)(17,20)(21,28)(23,31)(25,32)Polycyclic presentation:
2^9|c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330376);