Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16348 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,9)(3,8,15,21,13,23,20,24)(4,16,19,10,17,25,18,26)(14,28,30,32,27,31,29,22);(1,3,5,15,7,13,6,20)(2,8,11,21,12,23,9,24)(4,14,19,30,17,27,18,29)(10,22,25,28,26,32,16,31);(2,9)(3,13)(4,17)(5,6)(8,21)(10,25)(11,12)(16,26)(22,31)(23,24)(28,32)(29,30);(1,4,5,19,7,17,6,18)(2,10,11,25,12,26,9,16)(3,14,15,30,13,27,20,29)(8,22,21,28,23,32,24,31)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339668, #72339679, #72339684 |
| maximal normal subgroup of | #72339668, #72339679, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366953);