Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499914 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23918 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1380 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)(3,14)(4,13)(5,9)(6,10)(7,12)(8,15)(11,20)(16,30)(17,29)(18,25)(19,28)(21,24)(22,27)(23,32)(26,31);(1,3)(2,8)(4,18)(5,16)(6,15)(7,17)(9,26)(10,29)(11,24)(12,25)(13,14)(19,32)(20,31)(21,22)(23,27)(28,30);(2,9)(4,6)(5,20)(8,23)(10,28)(11,13)(12,27)(14,30)(15,18)(16,31)(19,22)(21,32)(24,29)(25,26);(1,3)(2,8)(4,18)(5,21)(6,15)(7,17)(9,23)(10,29)(11,30)(12,25)(13,14)(16,22)(19,31)(20,32)(24,28)(26,27)Polycyclic presentation:
2^9|e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,f^c=fh,h^c=hi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72214675 |
| maximal normal subgroup of | #72214675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407110);