Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,21,23,30)(9,24)(10,15)(11,14)(12,19)(13,22,29,25)(17,32,35,26)(18,28)(20,27)(31,33,36,34);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,26)(14,28)(15,27)(16,24)(21,31)(22,33)(23,35)(25,34)(29,32)(30,36);(2,4)(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,29)(15,30)(16,31)(18,33)(19,34)(20,35)(21,24)(27,36)(28,32);(6,14)(7,18)(9,10)(12,16)(13,29)(17,33)(19,20)(22,25)(26,31)(27,28)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237517, #72255889, #72255919, … |
| maximal normal subgroup of | #72237517, #72255889, #72255919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356684);