Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14)(7,11,17,27)(8,18,20,29)(9,22,24,13)(12,23,28,21)(15,30,19,35)(16,32,34,26)(25,33,36,31);(5,7)(6,11)(8,19,20,15)(9,16)(10,17)(12,25,28,36)(13,26)(14,27)(18,30,29,35)(21,33,23,31)(22,32)(24,34);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,29)(14,28)(16,33)(17,19)(18,22)(21,24)(26,35)(27,36)(30,32)(31,34);(1,2,3,4)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319067);