Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,21,27)(8,22,24,30)(9,15,10,14)(12,19,28,20)(13,25,29,23)(18,26,34,36)(31,33,35,32);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,27)(15,17)(16,28)(22,35)(23,32)(24,34)(25,33)(29,36)(30,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,30)(16,29)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36);(7,19)(9,10)(12,27)(14,15)(17,28)(18,32)(20,21)(22,30)(23,25)(26,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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 | #72250742, #72250771, #72250788, … |
| maximal normal subgroup of | #72250742, #72250771, #72250788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324902);