Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421042 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15283 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,17)(5,21)(9,26)(10,11)(13,24)(15,16)(19,20)(23,31)(27,29)(28,32);(1,2,6,11)(3,13)(4,18,20,22)(5,10,21,7)(8,15)(9,19,26,17)(12,25)(14,23)(16,24)(27,28)(29,31)(30,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,13)(9,23)(10,24)(11,25)(12,21)(17,27)(18,31)(19,29)(20,30)(22,28)(26,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,22)(8,23)(10,18)(11,26)(12,27)(13,28)(15,29)(16,30)(17,21)(24,31)(25,32)Polycyclic presentation:
2^9|b^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338551, #72338553, #72338560 |
| maximal normal subgroup of | #72338551, #72338553, #72338560 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328238);