Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423176 |
| 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 | |
1296 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)(4,18)(5,7)(9,22)(11,12)(13,25)(14,28)(15,16)(19,23)(20,21)(24,32)(29,30);(1,2)(3,13)(4,19)(5,12)(6,11)(7,8)(9,15)(10,20)(14,24)(16,25)(17,22)(18,26)(21,23)(27,29)(28,31)(30,32);(1,3,7,15,6,17,5,16)(2,9,12,13,11,25,8,22)(4,14,20,28,21,30,18,29)(10,24,19,27,26,32,23,31);(1,4)(2,10)(3,14)(5,18)(6,21)(7,20)(8,23)(9,24)(11,26)(12,19)(13,27)(15,28)(16,29)(17,30)(22,31)(25,32)Polycyclic presentation:
2^9|c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 330372);