Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,16)(8,20)(9,10)(14,29)(15,23)(18,19)(22,24)(25,27)(26,35)(31,33)(32,36);(1,2)(3,4)(5,6)(7,13)(8,21)(9,23)(10,12)(11,15)(14,20)(16,27)(17,32)(18,25)(19,28)(22,30)(24,29)(26,31)(33,36)(34,35);(2,4)(5,7)(6,13)(8,17)(9,16)(10,18)(11,19)(12,25)(14,26)(15,28)(20,31)(21,32)(22,34)(23,27)(24,33)(29,36)(30,35);(5,8,9,20,11,22,10,24)(6,14,12,21,15,29,23,30)(7,17,16,31,19,34,18,33)(13,26,25,32,28,36,27,35)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318430, #72318867, #72318957 |
| maximal normal subgroup of | #72318430, #72318867, #72318957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325700);