Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420401 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15277 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,8,7,12,5,11)(3,13,16,25,17,22,14,9)(4,18,19,23,21,10,20,26)(15,27,28,24,30,32,29,31);(2,8)(3,14)(4,19)(5,6)(11,12)(13,22)(16,17)(20,21)(23,26)(24,27)(28,29)(31,32);(1,3,5,16,7,17,6,14)(2,9,11,13,12,25,8,22)(4,15,19,29,21,30,20,28)(10,24,26,27,18,31,23,32);(1,4,5,19,7,21,6,20)(2,10,11,26,12,18,8,23)(3,15,16,29,17,30,14,28)(9,24,13,27,25,31,22,32)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216142, #72216192, … |
| maximal normal subgroup of | #72215936, #72216142, #72216192, … |
| maximal quotient of | #72237179, C2×#420401, #72252101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327597);