Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,6,11,22,13)(3,8,15,23,16,4,9,18)(5,10,21,27,20,28,24,29)(14,17,26,31,19,25,30,32);(1,3)(2,8,11,4)(5,19,20,14)(6,16)(7,15)(9,22)(10,17)(12,23,13,18)(21,30,24,26)(25,28)(27,31)(29,32);(1,4)(2,9)(3,13)(5,17)(6,8)(7,23)(10,26)(11,15)(12,16)(14,27)(18,22)(19,29)(20,25)(21,32)(24,31)(28,30);(1,5,6,20)(2,10)(3,14)(4,17,8,25)(7,24,22,21)(9,26)(11,28)(12,29)(13,27)(15,30)(16,19)(18,32,23,31)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338623, #72338625, #72338629 |
| maximal normal subgroup of | #72338623, #72338625, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343704);