Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500115 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24005 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60461 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,17)(5,6)(8,22)(9,24)(10,21)(13,25)(14,29)(15,30)(16,18)(19,20)(23,32)(26,31)(27,28);(1,2)(3,13)(4,9)(5,21)(6,10)(7,11)(8,15)(12,28)(14,27)(16,23)(17,24)(18,22)(19,26)(20,31)(25,29)(30,32);(1,3,6,15,7,14,5,16)(2,8,10,27,11,23,21,13)(4,18,17,29,19,30,20,12)(9,25,24,32,26,28,31,22);(4,19)(9,26)(12,29)(17,20)(18,30)(22,32)(24,31)(25,28)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi,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, 407311);