Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,16,8,14,11,27)(7,17,18,33,19,12,21,29)(9,23,25,31,22,13,26,15)(20,35,34,28,24,30,36,32);(2,3)(5,7,11,21,8,19,10,18)(6,12,27,33,14,17,16,29)(9,24,26,34,22,20,25,36)(13,30,31,28,23,35,15,32);(6,13)(7,18)(9,25)(10,11)(12,28)(14,23)(15,27)(16,31)(17,32)(19,21)(20,24)(22,26)(29,35)(30,33);(1,3,4,2)(7,19)(12,17)(18,21)(20,24)(28,32)(29,33)(30,35)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249916, #72250863, #72250872, … |
| maximal normal subgroup of | #72249916, #72250863, #72250872, … |
| maximal quotient of | #72287702, #72309434, #72309703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405532);