Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6462912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,10)(5,8)(6,11)(7,12)(9,16)(14,22)(15,27)(17,26)(18,21)(19,24)(20,25)(23,30)(28,31)(29,32);(2,8)(3,14)(4,18)(5,6)(9,22)(10,24)(11,12)(15,29)(16,17)(19,20)(23,32)(27,31);(1,3,6,16,7,17,5,14)(2,9,8,13,12,22,11,26)(4,15,20,30,19,29,18,28)(10,23,21,27,24,31,25,32);(1,4,5,18,7,19,6,20)(2,10,8,21,12,24,11,25)(3,15,14,28,17,29,16,30)(9,23,13,27,22,31,26,32);(4,19)(10,24)(15,29)(18,20)(21,25)(23,31)(27,32)(28,30)Polycyclic presentation:
2^9|c^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216037, #72216077, … |
| maximal normal subgroup of | #72215936, #72216037, #72216077, … |
| maximal quotient of | C2×#6462912, #72252171, #72255617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370108);